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

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The complete homogeneous symmetric polynomials $h_{1}, \dots, h_{n}$ freely generate the symmetric-polynomial ring

Statement

Substitution $U_{k} \mapsto h_{k}$ is an $R$ -algebra isomorphism

$R [U_{1}, \dots, U_{n}] ⟶ R [x_{1}, \dots, x_{n}]^{{Sym}_{n}} .$

Thus $h_{1}, \dots, h_{n}$ freely generate the symmetric-polynomial ring.

Facts & Assumptions

Given: A commutative ring $R$ and a natural number $n$ .

[L1]

The identity $E (- t) H (t) = 1$ gives $h_{k} - e_{1} h_{k - 1} + \dots + (- 1)^{k} e_{k} = 0$ for $1 \leq k \leq n$ (The generating-series identity $E (- t) H (t) = 1$ ).

[L2]

Substitution $T_{k} \mapsto e_{k}$ is an isomorphism from a polynomial ring onto the symmetric-polynomial ring (Fundamental theorem of symmetric polynomials: unique expression as a polynomial in $e_{1}, \dots, e_{n}$ ).

Proof

technique · direct

1.1L1algebra

The recurrence in [L1] expresses $h_{k}$ as $(- 1)^{k - 1} e_{k}$ plus a polynomial in $e_{1}, \dots, e_{k - 1}$ , and also expresses $e_{k}$ as $(- 1)^{k - 1} h_{k}$ plus a polynomial in $h_{1}, \dots, h_{k - 1}$ .

2.1step 1.1algebra

Recursion on $k$ therefore gives mutually inverse triangular substitutions between $R [e_{1}, \dots, e_{n}]$ and $R [h_{1}, \dots, h_{n}]$ ; every diagonal coefficient is $1$ or $- 1$ , hence a unit in $R$ .

3.1step 2.1L2∎

Composing either triangular isomorphism with [L2] shows that $U_{k} \mapsto h_{k}$ is an $R$ -algebra isomorphism onto the symmetric-polynomial ring.

Depends on

Used by

Complete homogeneous symmetric polynomials and their recurrence in two variables Example

Dependency tree · two levels

6 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)

The complete homogeneous symmetric polynomials h1,…,hnh1​,…,hn​ freely generate the symmetric-polynomial ring