TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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

Statement

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

$f (t) = \prod_{i = 1}^{n} (t - α_{i}), g (t) = \prod_{j = 1}^{m} (t - β_{j}),$

then

$Res (f, g) = \prod_{i = 1}^{n} \prod_{j = 1}^{m} (α_{i} - β_{j}) .$

If either degree is zero, both sides are the same empty product.

Facts & Assumptions

Given: Monic polynomials $f, g$ splitting in a common field extension as in the Statement.

[L1]

For monic $f$ with roots $α_{i}$ , the resultant satisfies $Res (f, g) = \prod_{i} g (α_{i})$ (For monic $f$ , $Res (f, g) = \prod_{i} g (α_{i})$ and it vanishes exactly when $f$ and $g$ have a common root).

[L2]

A split monic polynomial $g$ has the factorization $g (t) = \prod_{j} (t - β_{j})$ with roots counted with multiplicity (Polynomials that split and splitting fields of a polynomial or a family of polynomials).

Proof

technique · direct

1.1

Evaluate the factorization in [L2] at each $α_{i}$ to get $g (α_{i}) = \prod_{j = 1}^{m} (α_{i} - β_{j})$ .

givenL2

2.1

Substitute step 1.1 into [L1] and reassociate the finite product to obtain the double product.

step 1.1L1algebra

3.1

If $n = 0$ , [L1] is an empty outer product; if $m = 0$ , every $g (α_{i}) = 1$ and the inner products are empty. In either case both sides equal $1$ .

step 2.1L1L2∎

Depends on

Used by

For monic f,g of degrees n,m, Res(f,g)=(-1)ᵐⁿRes(g,f) Corollary

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 19 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, definition preceding Proposition 4.35 (standard reference, not scraped)

For monic f,gf,g of degrees n,mn,m splitting in a common extension, Res⁡(f,g)=∏i=1n∏j=1m(αi−βj)Res(f,g)=∏i=1n​∏j=1m​(αi​−βj​)