Theorem 1

Tag 06OG, permanent. Cite it as [Bourbaki, Tag 06OG].

Algebra, chapter V, § 11. Abelian extensions

Let $p$ be the characteristic exponent of K and let $n > 0$ be an integer. Then the group $\mu_n(K)$ of n-th roots of unity in K is cyclic and its order divides $n$ ; when K is algebraically closed and $n$ is prime to $p$ , the group $\mu_n(K)$ is cyclic of order $n$ .

It is enough to prove the first assertion of the theorem which is a consequence of the following more precise lemma:

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

Cited by

Lemma 1, § 9
Remark 3, § 11
Proposition 2, § 11
Proposition 3, § 5

Cites

Nothing.