TheoremStatement: Literature-sourcedProof: Literature-sourcedprecheck 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.

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Over an integral domain, degrees add under multiplication of nonzero polynomials

Statement

If $R$ is an integral domain and $f, g \in R [x]$ are nonzero, then $f g \neq 0$ and

$\deg (f g) = \deg f + \deg g, lc (f g) = lc (f) lc (g) .$

Facts & Assumptions

Given: An integral domain $R$ and nonzero polynomials $f, g \in R [x]$ .

[L1]

The coefficient of degree $\deg f + \deg g$ in $f g$ is the product of the two leading coefficients, and all higher coefficients vanish (Degree inequalities for sums and products over a commutative ring).

[L2]

In an integral domain, a product of two nonzero elements is nonzero (Zero divisor, and integral domain: a commutative ring with $1 \neq 0$ and no zero divisors).

Proof

technique · direct

1.1givenL2

The leading coefficients of $f$ and $g$ are nonzero, so [L2] makes their product nonzero.

2.1step 1.1L1∎

Fact [L1] identifies that product as the coefficient at degree $\deg f + \deg g$ and makes every higher coefficient zero, so $f g \neq 0$ , its degree is the sum of the degrees, and its leading coefficient is the product of the leading coefficients.

Over an integral domain, degrees add under multiplication of nonzero polynomials

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources