Lemma 1

Tag 0A6Q, permanent. Cite it as [Bourbaki, Tag 0A6Q].

Lie Groups and Lie Algebras, chapter II, § 8. Convergence of the Hausdorff series (ultrametric case)

Let $n$ be an integer $\geqslant 0$ and let $n = n_0 + n_1 p + \cdots + n_k p^k$ , with $0 \leqslant n_i \leqslant p - 1$ , be the $p$ -adic expansion of $n$ . Let $S(n) = n_0 + n_1 + \cdots + n_k$ . Then

v_p(n!) = \frac{n - S(n)}{p - 1}.

v_p(n!) = \sum_{i=1}^n v_p(i)

and the number of integers

i

between 1 and

n

for which

v_p(i) \geqslant j

is equal to the integral part

[n/p^j]

n/p^j

. Then

v_p(n!) = \sum_{j \geqslant 0} j ([n/p^j] - [n/p^{j+1}]) = \sum_{j \geqslant 1} [n/p^j].

[n/p^j] = \sum_{i \geqslant j} n_i p^{i-j}

, the lemma follows.

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

Cited by

Lemma 2, § 8
Proposition 14, § 7

Cites

Nothing.