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

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For monic $f$ of degree $n$ , $Res (f, f^{'}) = (- 1)^{n (n - 1) / 2} Disc (f)$

Statement

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

$Res (f, f^{'}) = (- 1)^{n (n - 1) / 2} Disc (f) .$

Facts & Assumptions

Given: A monic polynomial $f \in F [t]$ of degree $n$ , split as $f (t) = \prod_{i = 1}^{n} (t - α_{i})$ in a splitting field.

[L1]

The root-product formula gives $Res (f, f^{'}) = \prod_{i} f^{'} (α_{i})$ (For monic $f$ , $Res (f, g) = \prod_{i} g (α_{i})$ and it vanishes exactly when $f$ and $g$ have a common root).

[L2]

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

[L3]

The formal derivative of $f = \sum_{i} a_{i} x^{i}$ is $f^{'} = \sum_{i \geq 1} i a_{i} x^{i - 1}$ (The formal derivative of a polynomial).

[L4]

Over a commutative ring, formal differentiation is additive and $F$ -linear and satisfies $(f g)^{'} = f^{'} g + f g^{'}$ (Linearity, power rule, Leibniz rule and the degree bound for the formal derivative).

Proof

technique · direct

1.1givenL3L4algebra

Iterating the Leibniz rule of [L4] over the $n$ factors of $f = \prod_{i = 1}^{n} (t - α_{i})$ gives $f^{'} = \sum_{i = 1}^{n} \prod_{j \neq i} (t - α_{j})$ , since each $(t - α_{i})^{'} = 1$ by [L3]. At $t = α_{i}$ , every summand except the $i$ -th contains the zero factor $α_{i} - α_{i}$ , so $f^{'} (α_{i}) = \prod_{j \neq i} (α_{i} - α_{j})$ .

2.1step 1.1L1algebra

By [L1], $Res (f, f^{'}) = \prod_{i} \prod_{j \neq i} (α_{i} - α_{j})$ . Group the two ordered factors belonging to each unordered pair $i < j$ .

3.1step 2.1algebra

For each $i < j$ , $(α_{i} - α_{j}) (α_{j} - α_{i}) = - (α_{i} - α_{j})^{2}$ . There are $n (n - 1) / 2$ unordered pairs, so step 2.1 becomes $(- 1)^{n (n - 1) / 2} \prod_{i < j} (α_{i} - α_{j})^{2}$ .

4.1step 3.1L2∎

Apply [L2] to identify the remaining product with $Disc (f)$ . The empty-degree cases $n = 0, 1$ give $1 = 1$ .

Depends on

Used by

The discriminant of x³+px+q is -4p³-27q² Example

Dependency tree · two levels

13 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 and discriminant discussion (standard reference, not scraped)

For monic ff of degree nn, Res⁡(f,f′)=(−1)n(n−1)/2Disc⁡(f)Res(f,f′)=(−1)n(n−1)/2Disc(f)