DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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The discriminant of a monic polynomial as the coefficient expression of $Δ_{n}^{2}$

Definition

By The square of the Vandermonde polynomial is symmetric and Fundamental theorem of symmetric polynomials: unique expression as a polynomial in $e_{1}, \dots, e_{n}$ , there is a unique polynomial $D_{n} \in Z [T_{1}, \dots, T_{n}]$ such that

$Δ_{n} (x_{1}, \dots, x_{n})^{2} = D_{n} (e_{1}, \dots, e_{n}) .$

For a monic polynomial

$f (t) = t^{n} + a_{1} t^{n - 1} + \dots + a_{n}$

over a commutative ring, its discriminant is

$Disc (f) : = D_{n} (- a_{1}, a_{2}, \dots, (- 1)^{n} a_{n}) .$

Equivalently, in any algebra in which $f$ splits with roots $α_{1}, \dots, α_{n}$ , this coefficient expression evaluates to $Δ_{n} (α_{1}, \dots, α_{n})^{2}$ . The definition therefore depends only on the coefficients and not on a choice or ordering of roots. For a monic constant polynomial, $Disc (1) = 1$ .

Depends on

Used by

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

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 22 results over 9 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

J. S. Milne, Fields and Galois Theory, Proposition 4.35 through Example 4.37 (standard reference, not scraped)

The discriminant of a monic polynomial as the coefficient expression of Δn2Δn2​