LemmaStatement: Literature-sourcedProof: Literature-sourcedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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.

The monic gcd of two base-field polynomials is unchanged after extending the coefficient field

Statement

Let $F\subseteq E$ be a field extension and let $f,g\in F[x]$ be not both zero. The monic gcd of $f$ and $g$ computed in $F[x]$ is also their monic gcd in $E[x]$ .

Facts & Assumptions

Given: A subfield $F\subseteq E$ and polynomials $f,g\in F[x]$ not both zero.

[L1]

The monic gcd is the unique monic common divisor divisible by every common divisor (The monic greatest common divisor of two polynomials over a field).

[L2]

If $d=\gcd(f,g)$ in $F[x]$ , then $d=Af+Bg$ for some $A,B\in F[x]$ (Bézout identity and the Euclidean algorithm for polynomials over a field).

[L3]

A subfield has the same zero, one, operations, and inverses as the ambient field (Subfield: a subring of a field closed under inverses of its nonzero elements, and therefore a field with the restricted operations).

[L4]

A coefficient inclusion extends uniquely to a ring homomorphism of polynomial rings (Universal property of $R[x]$ : a coefficient homomorphism and the image of $x$ determine a unique ring homomorphism).

Proof

technique · direct

1.1

Let $d$ be the monic gcd in $F[x]$ ; it divides $f,g$ there and hence in $E[x]$ under [L4], while [L2] remains the identity $d=Af+Bg$ in $E[x]$ by [L3] and [L4].

givenL1L2L3L4

2.1

Every common divisor of $f,g$ in $E[x]$ divides the right side of the Bézout identity and hence divides $d$ ; since $d$ is monic, [L1] identifies it as the monic gcd computed in $E[x]$ .

step 1.1L1L2∎

Depends on

Used by

A nonzero polynomial over a field is separable exactly when its gcd with its derivative is 1 Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 50 results over 15 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

Brian Conrad, Differential Criterion and Primitivity, Lemma 1.1 (standard reference, not scraped)