四次方程的解（伪）

先在开头讲明了：本文的结果有可能有错，仅仅是作者打发时间副产品的记录．

解四次方程还是比解四次方程要难一些的．之前就看到有人高中的时候解代数方程解得很开心，就想自己也解一下．三次方程的解法在 Artin 的书里有，就没法自己解了，但是书里介绍四次方程伽罗瓦群的判定方法的时候提到了三次预解式 (resolvent cubic)，下午打发时间的时候就给弄出来了，但是好像跟网上的不大一样，就放在这里图一乐．

首先设出方程，并设解为 xi\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_i．这样设的话 sn\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}s_n 刚好就是 ∑xi1⋯xin\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 x_{i_1}\cdots x_{i_n}，比较方便： f(x)=x4+s2x2−s3x+s4=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}f(x)=x^4+s_2x^2-s_3x+s_4=0 这里设 s1=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}s_1=0，能简化后面的式子．

定义 b1=x1x2+x3x4,b2=x1x3+x2x4,b3=x1x4+x2x3\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_1=x_1x_2+x_3x_4,\\ b_2=x_1x_3+x_2x_4,\\ b_3=x_1x_4+x_2x_3 b\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 就会是三次方程的解，而且系数可以用 s\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}s 表示：

g(x)=(x−b1)(x−b2)(x−b3)=x3−s2x2−4s4x−s32+4s2s4\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(x)=(x-b_1)(x-b_2)(x-b_3)=x^3-s_2x^2-4s_4x-s_3^2+4s_2s_4 所以我们可以用三次方程的求根公式解出 b\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.

再设 c1=x1x2−x3x4,c2=x1x3−x2x4,c3=x1x4−x2x3\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_1=x_1x_2-x_3x_4,\\ c_2=x_1x_3-x_2x_4,\\ c_3=x_1x_4-x_2x_3 就有 bi2−ci2=4s4\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_i^2-c_i^2=4s_4 所以我们就解出了 c\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.

考虑到 (x2+x3+x4)2=x22+x32+x42+2x2x3+2x2x4+2x3x4\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_2+x_3+x_4)^2=x_2^2+x_3^2+x_4^2+2x_2x_3+2x_2x_4+2x_3x_4 而 x2x3=b3−c32,x2x4=b2−c22,x3x4=b1−c12,x12+x22+x32+x42=s12−2s2\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_2x_3=\frac{b_3-c_3}{2},\\ x_2x_4=\frac{b_2-c_2}{2},\\ x_3x_4=\frac{b_1-c_1}{2},\\ x_1^2+x_2^2+x_3^2+x_4^2=s_1^2-2s_2 就有 x12=(s1−x1)2=−x12+s12−2s2+s2−c1−c2−c3,x1=−s2−b12−4s4−b22−4s4−b32−4s42\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_1^2=(s_1-x_1)^2=-x_1^2+s_1^2-2s_2+s_2-c_1-c_2-c_3,\\ x_1=\sqrt{\frac{-s_2-\sqrt{b_1^2-4s_4}-\sqrt{b_2^2-4s_4}-\sqrt{b_3^2-4s_4}}{2}} 这里根号并不表示取哪一个值，因为解出一坨根式本身就没有很大的价值．虽然和网上的不一样，我用 x4−15x2−10x+24\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^4-15 x^2-10 x+24 来测试的时候还是可以得出解的．我的话就只能把每个立方根和平方根都试一遍，确实能得出所有解．

这里解出来 x4+bx2−cx+d=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}x^4+bx^2-cx+d=0 的解就是（溢出了） x=−b−(2b3+(2b3−72bd+27c2)2+4(−b2−12d)3−72bd+27c23323−23(−b2−12d)32b3+(2b3−72bd+27c2)2+4(−b2−12d)3−72bd+27c23+b3)2−4d−(2b3+(2b3−72bd+27c2)2+4(−b2−12d)3−72bd+27c23323−23(−b2−12d)32b3+(2b3−72bd+27c2)2+4(−b2−12d)3−72bd+27c23+b3)2−4d−(2b3+(2b3−72bd+27c2)2+4(−b2−12d)3−72bd+27c23323−23(−b2−12d)32b3+(2b3−72bd+27c2)2+4(−b2−12d)3−72bd+27c23+b3)2−4d2.\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=\sqrt{\frac{-b-\sqrt{\left(\frac{\sqrt[3]{2 b^3+\sqrt{\left(2 b^3-72 b d+27 c^2\right)^2+4 \left(-b^2-12 d\right)^3}-72 b d+27 c^2}}{3 \sqrt[3]{2}}-\frac{\sqrt[3]{2} \left(-b^2-12 d\right)}{3 \sqrt[3]{2 b^3+\sqrt{\left(2 b^3-72 b d+27 c^2\right)^2+4 \left(-b^2-12 d\right)^3}-72 b d+27 c^2}}+\frac{b}{3}\right)^2-4 d}-\sqrt{\left(\frac{\sqrt[3]{2 b^3+\sqrt{\left(2 b^3-72 b d+27 c^2\right)^2+4 \left(-b^2-12 d\right)^3}-72 b d+27 c^2}}{3 \sqrt[3]{2}}-\frac{\sqrt[3]{2} \left(-b^2-12 d\right)}{3 \sqrt[3]{2 b^3+\sqrt{\left(2 b^3-72 b d+27 c^2\right)^2+4 \left(-b^2-12 d\right)^3}-72 b d+27 c^2}}+\frac{b}{3}\right)^2-4 d}-\sqrt{\left(\frac{\sqrt[3]{2 b^3+\sqrt{\left(2 b^3-72 b d+27 c^2\right)^2+4 \left(-b^2-12 d\right)^3}-72 b d+27 c^2}}{3 \sqrt[3]{2}}-\frac{\sqrt[3]{2} \left(-b^2-12 d\right)}{3 \sqrt[3]{2 b^3+\sqrt{\left(2 b^3-72 b d+27 c^2\right)^2+4 \left(-b^2-12 d\right)^3}-72 b d+27 c^2}}+\frac{b}{3}\right)^2-4 d}}{2}}.