fundamental theorem of symmetric functions

Some polynomial expressions in several variables don't care about the order of those variables — swapping any two leaves the expression unchanged. The fundamental theorem of symmetric functions says that every such expression, no matter how complicated, is built in a unique way out of a short standard list of basic symmetric building blocks.

Precisely, a polynomial in x_1, ..., x_n is symmetric if it is invariant under every permutation of the variables. The elementary symmetric polynomials e_1 = sum x_i, e_2 = sum_{i<j} x_i x_j, ..., e_n = x_1 x_2 ... x_n are the basic blocks. The theorem states that every symmetric polynomial can be written as a polynomial in e_1, ..., e_n, with coefficients in the base ring, and this expression is unique. In ring language, the ring of symmetric polynomials equals the polynomial ring R[e_1, ..., e_n], and the e_i are algebraically independent.

This connects directly to roots and coefficients: if r_1, ..., r_n are the roots of a monic polynomial, then up to signs the e_i(r_1, ..., r_n) are exactly its coefficients (Vieta's formulas). Therefore any symmetric expression in the roots — the discriminant, power sums, resolvent coefficients — is automatically a polynomial in the coefficients, hence lies in the base field. This is the precise reason Galois-theoretic invariants are computable without ever leaving the ground field.