Lemma 3

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

Commutative Algebra, chapter VII, § 3. Factorial domains

Let $f$ be a non-zero element of $E[[X_1, \ldots, X_n]]$ . There exist integers $u(i) \geq 1 (1 \leq i \leq n-1)$ such that the automorphism $s$ of $E$ defined by

s(X_i) = X_i + X_n^{u(i)} \quad (1 \leq i \leq n-1)

and $s(X_n) = X_n$ , maps $f$ to an element $g$ such that $g(0, \ldots, 0, X_n) \neq 0$ .

$g(0, \ldots, 0, X_n) = f(X_n^{u(1)}, \ldots, X_n^{u(n-1)}, X_n)$ . Lemma 3 is therefore a consequence of Lemma 2.

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

Cited by

Proposition 8, § 3

Cites

Lemma 2