Lemma 1 (Gauss)

Tag 051S, permanent. Cite it as [Bourbaki, Tag 051S].

Commutative Algebra, chapter VII, § 3. Factorial domains

Let $f, f'$ be non-zero elements of $K[X]$ and c, $c'$ contents off, $f', f'$ . Then $cc'$ is a content of $ff'$ .

Let d be a content of $ff'$ . For every valuation $\nu$ on K which is essential for A, let $\bar{\nu}$ denote its canonical extension to $K[X]$ . Then

\nu(d) = \bar{\nu}(ff') = \bar{\nu}(f) + \bar{\nu}(f') = \nu(c) + \nu(c') = \nu(cc').

Hence

cc'd^{-1}

is an invertible element of A.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Theorem 2, § 3

Cites

Nothing.