Symmetric Polynomials and the Fundamental Theorem of Symmetric Functions

About this page

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

✓ 21 results · all verified · 12 also independently AI-judged

Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 9 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.

1 · Prerequisites

Abstract Algebra

Combinatorics

Finite Counting, Factorials and Binomial Coefficients

Real Analysis

Set Theory

2 · Logical flowchart

3 · Items

Definition

Symmetric Polynomials and the Fundamental Theorem of Symmetric Functions

3 · Items

Symmetric polynomials as the invariants of variable permutations

The symmetric polynomials form a subring

The elementary symmetric polynomials e₀,e₁,…,eₙ

Vieta expansion: ∏ᵢ₌₁ⁿ(t-xᵢ)=∑ₖ₌₀ⁿ(-1)ᵏ eₖ tⁿ⁻ᵏ

Vieta's formulas identify the coefficients of a split monic polynomial with elementary symmetric functions of its roots

Monomial symmetric polynomials indexed by partitions

Monomial symmetric polynomials form an R-basis of the symmetric-polynomial ring

Lexicographic order on exponent tuples and the multidegree of a nonzero polynomial

The leading multidegree of a symmetric polynomial is weakly decreasing

The leading multidegree of e₁^b₁⋯ eₙ^bₙ is (b₁+⋯+bₙ,b₂+⋯+bₙ,…,bₙ) with coefficient one

Every symmetric polynomial over a commutative ring is a polynomial in the elementary symmetric polynomials

The elementary symmetric polynomials are algebraically independent over the coefficient ring

Fundamental theorem of symmetric polynomials: unique expression as a polynomial in e₁,…,eₙ

A symmetric polynomial in the roots of a monic polynomial is a polynomial in its coefficients and lies in the base ring

Power sums pₖ and complete homogeneous symmetric polynomials hₖ

The generating-series identity E(-t)H(t)=1

The complete homogeneous symmetric polynomials h₁,…,hₙ freely generate the symmetric-polynomial ring

Newton's identities: k eₖ=∑ᵢ₌₁ᵏ(-1)ⁱ⁻¹eₖ₋ᵢpᵢ

If n! is invertible, then p₁,…,pₙ freely generate the symmetric-polynomial ring

The Vandermonde polynomial Δₙ=∏_i<j(xᵢ-xⱼ)

The square of the Vandermonde polynomial is symmetric

The discriminant of a monic polynomial as the coefficient expression of Δₙ²

The discriminant is ∏_i<j(αᵢ-αⱼ)² and vanishes exactly when a monic polynomial has a repeated root

The monic resultant Res(f,g) from the symmetric coefficient expression of ∏ᵢ g(xᵢ)

For monic f, Res(f,g)=∏ᵢ g(αᵢ) and it vanishes exactly when f and g have a common root

For monic f,g of degrees n,m splitting in a common extension, Res(f,g)=∏ᵢ₌₁ⁿ∏ⱼ₌₁ᵐ(αᵢ-βⱼ)

For monic f,g of degrees n,m, Res(f,g)=(-1)ᵐⁿRes(g,f)

For monic f, if g-g₁=qf, then Res(f,g)=Res(f,g₁)

For monic f of degree n, Res(f,f')=(-1)^n(n-1)/2Disc(f)