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单词 GaloistheoreticDerivationOfTheCubicFormula
释义

Galois-theoretic derivation of the cubic formula


We are trying to find the roots r1,r2,r3 of the polynomialPlanetmathPlanetmath x3+ax2+bx+c=0. From the equation

(x-r1)(x-r2)(x-r3)=x3+ax2+bx+c

we see that

a=-(r1+r2+r3)
b=r1r2+r1r3+r2r3
c=-r1r2r3

The goal is to explicitly construct a radical tower over the fieldk=(a,b,c) that contains the three roots r1,r2,r3.

Let L=(r1,r2,r3). By Galois theoryMathworldPlanetmath we know thatGal(L/(a,b,c))=S3. Let KL be the fixed field ofA3S3. We have a tower of field extensions

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更新时间:2025/5/4 20:06:44