CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck 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

1.1

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 $h_{1}, \dots, h_{k - 1}$ .

L1algebra

2.1

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 $R$ .

step 1.1algebra

3.1

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

step 2.1L2∎

Depends on

Used by

Complete homogeneous symmetric polynomials and their recurrence in two variables Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 12 results over 6 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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