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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

1.1

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

givenL1L2

2.1

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.

step 1.1algebra

3.1

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.

step 2.1L2L3∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 28 results over 7 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 is ∏i<j(αi−αj)2∏i<j​(αi​−αj​)2 and vanishes exactly when a monic polynomial has a repeated root