TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-16

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The discriminant is $\prod_{i < j} (α_{i} - α_{j})^{2}$ and vanishes exactly when a monic polynomial has a repeated root

Statement

Let $F$ be a field and let $f \in F [t]$ be monic of degree $n$ . In a splitting field write

$f (t) = \prod_{i = 1}^{n} (t - α_{i}) .$

Then

$Disc (f) = \prod_{1 \leq i < j \leq n} (α_{i} - α_{j})^{2} .$

Moreover, $Disc (f) = 0$ if and only if $f$ has a repeated root. This criterion holds in every characteristic.

Facts & Assumptions

Given: A field $F$ , a monic polynomial $f$ , and a splitting field with roots $α_{1}, \dots, α_{n}$ .

[L1]

The discriminant is the coefficient expression obtained from $Δ_{n}^{2}$ , and in a split algebra it evaluates to $Δ_{n} (α_{1}, \dots, α_{n})^{2}$ (The discriminant of a monic polynomial as the coefficient expression of $Δ_{n}^{2}$ ).

[L2]

A splitting field presents $f$ as a product of linear factors with roots counted according to multiplicity (Polynomials that split and splitting fields of a polynomial or a family of polynomials).

[L3]

A root $a$ of a nonzero polynomial is repeated if and only if $f^{'} (a) = 0$ (A root is repeated exactly when it is also a root of the formal derivative).

Proof

technique · direct

1.1givenL1L2

Evaluate the coefficient expression in [L1] at the roots supplied by [L2]. The definition of $Δ_{n}$ gives $Disc (f) = \prod_{i < j} (α_{i} - α_{j})^{2}$ .

2.1step 1.1algebra

Because the splitting field is a field, this finite product is zero exactly when one factor $α_{i} - α_{j}$ is zero, equivalently when two entries in the root list coincide.

3.1step 2.1L2L3∎

Two entries coincide exactly when the linear factor at that root occurs at least twice, so $f$ has a repeated root. By [L3] this agrees with the derivative criterion. No step divides by $2$ , so the equivalence remains valid in characteristic two.

Depends on

Used by

Dependency tree · two levels

12 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

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

The discriminant is ∏i<j(αi−αj)2∏i<j​(αi​−αj​)2 and vanishes exactly when a monic polynomial has a repeated root