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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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

1.1givenL2

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

2.1step 1.1L1algebra

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

3.1step 2.1L1L2∎

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 $1$ .

Depends on

Used by

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

Dependency tree · two levels

8 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, 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​)