Proposition 12

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

Suppose that $p \neq 0$ . Let $x \in G_f$ . The following conditions are equivalent: (i) $\log x = 0$ ; (ii) $x$ is of finite order in $G$ .

If there exists an integer $n > 0$ such that $x^n = e$ , it follows that

n \log x = \log x^n = 0,

whence

\log x = 0

. If

\log x = 0

, let

V

be a neighbourhood of

e

G_f

such that

\log|V|

is the inverse mapping of an injective exponential mapping. There exists an integer

n > 0

such that

x^n \in V

; the equality

\log x^n = 0

implies

x^n = e

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

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