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Fundamental theorem of symmetric polynomials: unique expression as a polynomial in
Statement
For every commutative ring and every , substitution is an -algebra isomorphism
Equivalently, every symmetric polynomial has a unique expression .
Facts & Assumptions
Given: A commutative ring and a natural number .
Every symmetric polynomial is for some polynomial (Every symmetric polynomial over a commutative ring is a polynomial in the elementary symmetric polynomials).
The elementary symmetric polynomials are algebraically independent: implies (The elementary symmetric polynomials are algebraically independent over the coefficient ring).
Proof
Substitution defines an -algebra homomorphism whose image lies in the symmetric-polynomial subring.
The map is surjective by [L1].
Its kernel is zero by [L2], so it is injective.
The substitution map is therefore an isomorphism. Surjectivity gives existence of an expression, and injectivity gives its uniqueness.
Depends on
Used by
- A symmetric polynomial in the roots of a monic polynomial is a polynomial in its coefficients and lies in the base ring Corollary
- If n! is invertible, then p₁,…,pₙ freely generate the symmetric-polynomial ring Corollary
- The complete homogeneous symmetric polynomials h₁,…,hₙ freely generate the symmetric-polynomial ring Corollary
- Chern character of a complex vector bundle Definition
- The discriminant of a monic polynomial as the coefficient expression of Δₙ² Definition
- The monic resultant Res(f,g) from the symmetric coefficient expression of ∏ᵢ g(xᵢ) Definition
- The rank-one Soergel category Example
- The symmetric polynomials as the invariant ring of the symmetric group, seen through Noether's finiteness theorem Example
- The type A₂ rank-two Soergel decomposition Example
- The coefficient formula and discriminant of the quartic resolvent Proposition
- Integral cohomology of BU(n) Theorem
- The general polynomial of degree n has Galois group Sₙ Theorem
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
- K. Conrad, Symmetric Polynomials, Section 2 (standard reference, not scraped)
- D. Grinberg, An Introduction to Algebraic Combinatorics, Theorem 7.1.16 (standard reference, not scraped)