LemmaStatement: Literature-sourcedProof: Literature-sourcedprecheck passaudited 2026-08-11

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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 $d = A f + B g$ 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 $d = A f + B g$ 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

Dependency tree · two levels

21 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

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