CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 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 $f$ , if $g - g_{1} = q f$ , then $Res (f, g) = Res (f, g_{1})$

Statement

Let $F$ be a field, let $f \in F [t]$ be monic, and let $g, g_{1}, q \in F [t]$ satisfy

$g - g_{1} = q f .$

Then

$Res (f, g) = Res (f, g_{1}) .$

Facts & Assumptions

Given: Polynomials $f, g, g_{1}, q$ satisfying the identity in the Statement, with $f$ monic.

[L1]

If $f$ has roots $α_{i}$ in a splitting field, then $Res (f, h) = \prod_{i} h (α_{i})$ for every polynomial $h$ (For monic $f$ , $Res (f, g) = \prod_{i} g (α_{i})$ and it vanishes exactly when $f$ and $g$ have a common root).

[L2]

Polynomial evaluation is a ring homomorphism and an element $a$ is a root of $f$ exactly when $f (a) = 0$ (Evaluation and roots of a polynomial in a commutative target ring).

Proof

technique · direct

1.1givenL2algebra

For every root $α_{i}$ of $f$ , evaluate $g - g_{1} = q f$ to obtain $g (α_{i}) - g_{1} (α_{i}) = q (α_{i}) f (α_{i}) = 0$ , hence $g (α_{i}) = g_{1} (α_{i})$ .

2.1step 1.1L1

Apply [L1] to $g$ and $g_{1}$ and multiply the equal values from step 1.1 to obtain equality of the resultants.

3.1L1∎

If $\deg f = 0$ , then $f = 1$ and both resultants are the empty product $1$ ; the argument remains valid.

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(c) (standard reference, not scraped)

For monic ff, if g−g1=qfg−g1​=qf, then Res⁡(f,g)=Res⁡(f,g1)Res(f,g)=Res(f,g1​)