Symmetric Polynomials and Splitting Fields

The more symmetric is, the closer it is to the base field .

Symmetric Functions Theorem

Every symmetric polynomial with coefficients in a ring can be written in a unique way as a polynomial in the elementary symmetric polynomials.

e.g.

Corollary

Suppose that a polynomial has coefficients in a field , and it splits completely in an extension field , with roots . Let be a symmetric polynomial in with coefficients in . Then .

Proof Symmetric functions theorem tells that is a polynomial in the elementary symmetric functions. Say that , where as well. When we evaluate at , we obtain , so must also be in .

Splitting Fields

Splitting Theorem

Let be a splitting field extension of a polynomial . If an irreducible polynomial has one root in , then it splits completely in .

The Discriminant