Rings, Modules, Fields

16 Symmetric Polynomials

The symmetric polynomials have been behind the scenes so far in Galois Theory. We will turn to their study.

Definition 16.1. The elementary symmetric polynomials in the variables \(x_1,\dots ,x_n\) are the homogeneous polynomials \(\sigma _{i,n}\) (often shortened to \(\sigma _i\)) where \(\sigma _{i,n}\) is the \(i^{th}\) graded part of \(\Pi _1^n (t-x_i)\).

Note these polynomials are invariant under the \(S_n\) action that permutes the \(x_i\). We say that a symmetric polynomial is one that is invariant under this \(S_n\) action.

Theorem 16.2. Any symmetric polynomial in \(n\) variables is a polynomial in the \(\sigma _i\). In other words, given \(R[x_1,\dots ,x_n]\) the ring fixed by the \(S_n\) action is \(R[\sigma _1,\dots ,\sigma _n]\).

Proof. We can prove this directly via a division algorithm. We put an ordering on monomials \(\Pi _i x_i^{a_i}\) such that \((a_1,\dots ,a_n) > (b_1,\dots ,b_n)\) if the first term where they differ, \(a_i>b_i\). Now given a symmetric polynomial \(f\), suppose the leading monomial is \((a_1,\dots ,a_n)\) with coefficient \(r\), then \(f-r\sigma _1^{a_1-a_2}\sigma _2^{a_2-a_3}\dots \sigma _n^{a_n}\) has strictly smaller leading monomial. □

We can now consider the generic Galois extension, \(K(x_1,\dots ,x_n)/K(\sigma _1,\dots ,\sigma _n)\). The action of \(S_n\) by the previous theorem leaves \(K(\sigma _1,\dots ,\sigma _n)\) fixed, so the Galois group is \(S_n\). The minimal polynomial of the \(x_i\) is \(\prod _i(t-x_i) = \sum _i t^i\sigma _i\), so is degree \(n\)

\(S_n\) is solvable for \(n\leq 4\), so we can generally solve a degree \(4\) or less polynomial using radicals. Note that

Given a Galois element, an intermediate subextension, one can compute the minimal polynomial of the trace to the intermediate extension