12/04/15 21:35:27.96
>>41
>”（１）代数方程式は隠れた対称性をもっている。この対称性はガロア群*3で記述される。”
>この視点が気に入った
補足
例えば、５次方程式f(x)=x^5+ax^4+bx^3+cx^2+dx+e の５つ根α、β、γ、δ、ε
根と係数の関係から、係数は根の基本対称式になる
ここに隠れた対称性といわれる所以がある
URLﾘﾝｸ(en.wikipedia.org)
Symmetric polynomial
Galois theory
One context in which symmetric polynomial functions occur is in the study of monic univariate polynomials of degree n having n roots in a given field.
These n roots determine the polynomial, and when they are considered as independent variables, the coefficients of the polynomial are symmetric polynomial functions of the roots.
Moreover the fundamental theorem of symmetric polynomials implies
that a polynomial function f of the n roots can be expressed as (another) polynomial function of the coefficients of the polynomial determined by the roots if and only if f is given by a symmetric polynomial.
This yields the approach to solving polynomial equations in terms of inverting this map, "breaking" the symmetry ? given the coefficients of the polynomial (the elementary symmetric polynomials in the roots), how can one recover the roots?
This leads to studying solutions of polynomials in terms of the permutation group of the roots, originally in the form of Lagrange resolvents, later developed in Galois theory.