$\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\cos(2\pi/17)$

人真的可以很无聊．本来想着复习一下高考的，一不留神又打开了数论书，一不留神就开始算 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\cos(2\pi/17)$ 了，结果快整个上午都没了．虽然 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\zeta_{17}$ 可以用尺规作出很好理解，在计算的过程中还是遇到了一些困难．

先复习一下：若 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}p$ 是奇质数，$\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\zeta_p$ 在 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\Q$ 上的最小多项式是 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}x^{p-1}+x^{p-2}+\cdots+1$，$\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\Gal(\Q(\zeta_p)/\Q)\approx\Z_p^\times\approx C_{p-1}$，而尺规可以做的域扩张都是二次扩域．如果可以作出 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}p$ 边形，就要求在 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}C_{p-1}$ 里能找到从 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\{1\}$ 开始，到 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}C_{p-1}$ 的子群链，使得下一个子群除以上一个子群是 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}C_2$．这意味着 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}p-1=2^k$．

1796 年高斯就发现了正十七边形可以用尺规作出．十七边形是最小的比较难作的正质数边形，所以用来打发时间的话是个比较好的选择．

为了计算出 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\cos(2\pi/17)$，我们要选择 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\Gal(\Q(\zeta_{17})/\Q)$ 里一条合适的中间域链以及合适的生成元．模 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}17$ 的一个原根是 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}g=3$．如果用 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\Z_p^\times$ 的子群表示，我们要从 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\Q$ 对应的 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\{g^k\}$ 开始，扩展到 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\{g^{2k}\}$，再到 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\{g^{4k}\}$ 和 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\{g^{8k}\}$．因为 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\cos(2\pi/17)=(\zeta+\zeta^{-1})/2$，到这一步就可以了．生成元很好找，我选的是 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\zeta$ 在所有的自同构下像的和．下面为了简便，用 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}[a_1, a_2, \cdots, a_n]$ 代表 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\sum \zeta^{a_i}$．

为了求出 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}A, B, C$，我们要求出它们在前一个域上的最小多项式．我们只要求出它们的平方，然后消元就可以了．要注意的是求 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}B$ 或 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}C$ 的时候多项式的系数就可以不是有理数了．我的方法是在求 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}B$ 的时候不止把 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}B^2, B$ 表示出来，而且要表示出 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}AB^2, AB$ 表示出来．按理来讲求 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}C$ 的时候也需要这样，而且要计算 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}B^3$，但非常幸运，我只算了两个式子就可以消去不在系数域里的数了．最后找到的方程如下：

$$\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}A^2+A-4=0,$$ $$\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}(A+4)B^2-(3A+4)B-(A+4)=0$$ $$\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}2C^2-2BC+(B^2+B-A-4)=0$$ $$\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}C=2\cos\frac{2\pi}{17}$$

所以 $$\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\cos\frac{2\pi}{17}=\frac{\sqrt{17}-1+\sqrt{34-2\sqrt{17}}+\sqrt{4(3\sqrt{17}+17)-(\sqrt{17}+3)\sqrt{8(17-\sqrt{17})}}}{16}.$$