CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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For monic $f, g$ of degrees $n, m$ , $Res (f, g) = (- 1)^{m n} Res (g, f)$

Statement

Let $F$ be a field and let $f, g \in F [t]$ be monic of degrees $n, m$ . Then

$Res (f, g) = (- 1)^{m n} Res (g, f) .$

Facts & Assumptions

Given: Monic polynomials $f, g \in F [t]$ of degrees $n, m$ .

[L1]

In a common splitting extension, $Res (f, g) = \prod_{i, j} (α_{i} - β_{j})$ and $Res (g, f) = \prod_{j, i} (β_{j} - α_{i})$ (For monic $f, g$ of degrees $n, m$ splitting in a common extension, $Res (f, g) = \prod_{i = 1}^{n} \prod_{j = 1}^{m} (α_{i} - β_{j})$ ).

[L2]

Every nonzero polynomial over a field has a splitting field (Every nonzero polynomial over a field has a splitting field).

Proof

technique · direct

1.1givenL2

Take a splitting field of the nonzero polynomial $f g$ when both degrees are positive; if one polynomial is $1$ , use any splitting field for the other.

2.1step 1.1L1algebra

Apply [L1] in that field. Replacing each of the $m n$ factors $α_{i} - β_{j}$ by $- (β_{j} - α_{i})$ contributes the factor $(- 1)^{m n}$ and yields the formula.

3.1L1algebra∎

If $n = 0$ or $m = 0$ , both resultants are $1$ and $(- 1)^{m n} = 1$ , so the same identity holds.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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(a) (standard reference, not scraped)

For monic f,gf,g of degrees n,mn,m, Res⁡(f,g)=(−1)mnRes⁡(g,f)Res(f,g)=(−1)mnRes(g,f)