PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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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.

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The symmetric polynomials form a subring

Statement

For every commutative ring $R$ and every $n \in N$ , the symmetric polynomials $R [x_{1}, \dots, x_{n}]^{{Sym}_{n}}$ form a subring of $R [x_{1}, \dots, x_{n}]$ . For $n = 0$ this subring is $R$ .

Facts & Assumptions

Given: A commutative ring $R$ , a natural number $n$ , and symmetric polynomials $f, g \in R [x_{1}, \dots, x_{n}]$ .

[L1]

A polynomial is symmetric when every variable permutation fixes it (Symmetric polynomials as the invariants of variable permutations).

[L2]

A subset is a subring when it contains $1$ and is closed under addition, additive inverses, and multiplication (Subring: a subset containing $1_{R}$ and closed under addition, additive inverses and multiplication).

Proof

technique · direct

1.1L1

Every variable permutation fixes the constant polynomials $0$ and $1$ , so both are symmetric.

1.2givenL1algebra

For every permutation $σ$ , substitution of permuted variables commutes with the ring operations, so $σ (f + g) = σ (f) + σ (g) = f + g$ , $σ (- f) = - σ (f) = - f$ , and $σ (f g) = σ (f) σ (g) = f g$ .

2.1step 1.1step 1.2L2∎

Thus the symmetric polynomials contain $1$ and are closed under addition, additive inverses, and multiplication, so they form a subring. When $n = 0$ every polynomial is a coefficient in $R$ and the assertion gives $R$ itself.

Depends on

Used by

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

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

D. Grinberg, An Introduction to Algebraic Combinatorics, Chapter 7, Section 7.1 (standard reference, not scraped)