Let $K$ be a totally real (finite) number field. Let $S$ be a finite set of places of $K$ containing the primes above a prime number $p$. Let $K_S$ be the maximal abelian extension unramified outside $S$. Let $S_\infty$ the set of infinite places of $K$.

I have noticed the following terminology that is confusing me. Some french mathematicians (Serre, Colmez $\S1.2$) immediately claim that $K_S$ contains all the $p^n$-th roots of unity, for all $n\ge1$. This is clear if $S$ contains $S_\infty$. But, if $S$ does not contain any place in $S_\infty$, then $K_S$ is unramified at infinity, and is totally real, hence can't contain this roots of unity.

On the other hand, in articles by Ribet, for example, it is always mentioned if $S$ contains $S_\infty$ or not, thus there is no confusion.

Is it standard to assume that $S$ contains the infinite places, or one simply has to understand this from the context?

The other possibility is that I'm not understanding something here. Sorry if this is the case, I'm not fluent in class field theory.

Let $K$ be a totally real (finite) number field. Let $S$ be a finite set of places of $K$ containing the primes above a prime number $p$. Let $K_S$ be the maximal abelian extension unramified outside $S$. Let $S_\infty$ the set of infinite places of $K$.

I have noticed the following terminology that is confusing me. In some articles (e.g., Colmez $\S1.2$ and Serre $\S2.1$) immediately claim that $K_S$ contains all the $p^n$-th roots of unity, for all $n\ge1$. This is clear if $S$ contains $S_\infty$. But, if $S$ does not contain any place in $S_\infty$, then $K_S$ is unramified at infinity, and is totally real, hence can't contain this roots of unity.

On the other hand, in articles by Ribet, for example, it is always mentioned if $S$ contains $S_\infty$ or not, thus there is no confusion.

Is it standard to assume that $S$ contains the infinite places, or one simply has to understand this from the context?

The other possibility is that I'm not understanding something here. Sorry if this is the case, I'm not fluent in class field theory.

References:

Colmez, Pierre. Fonctions L p-adiques. Séminaire Bourbaki, Vol. 1998/99. Astérisque No. 266 (2000), Exp. No. 851, 3, 21–58.

Serre, Jean-Pierre. Sur le résidu de la fonction zêta p-adique d'un corps de nombres. C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 4, A183–A188.

Let $K$ be a totally real (finite) number field. Let $S$ be a finite set of places of $K$ containing the primes above a prime number $p$. Let $K_S$ be the maximal abelian extension unramified outside $S$. Let $S_\infty$ the set of infinite places of $K$.

I have noticed the following terminology that is confusing me. Some french mathematicians (Serre, Colmez) immediately claim that $K_S$ contains all the $p^n$-th roots of unity, for all $n\ge1$. This is clear if $S$ contains $S_\infty$. But, if $S$ does not contain any place in $S_\infty$, then $K_S$ is unramified at infinity, and is totally real, hence can't contain this roots of unity.

On the other hand, in articles by Ribet, for example, it is always mentioned if $S$ contains $S_\infty$ or not, thus there is no confusion.

Is it standard to assume that $S$ contains the infinite places, or one simply has to understand this from the context?

The other possibility is that I'm not understanding something here. Sorry if this is the case, I'm not fluent in class field theory.

Let $K$ be a totally real (finite) number field. Let $S$ be a finite set of places of $K$ containing the primes above a prime number $p$. Let $K_S$ be the maximal abelian extension unramified outside $S$. Let $S_\infty$ the set of infinite places of $K$.

I have noticed the following terminology that is confusing me. Some french mathematicians (Serre, Colmez $\S1.2$) immediately claim that $K_S$ contains all the $p^n$-th roots of unity, for all $n\ge1$. This is clear if $S$ contains $S_\infty$. But, if $S$ does not contain any place in $S_\infty$, then $K_S$ is unramified at infinity, and is totally real, hence can't contain this roots of unity.

On the other hand, in articles by Ribet, for example, it is always mentioned if $S$ contains $S_\infty$ or not, thus there is no confusion.

Is it standard to assume that $S$ contains the infinite places, or one simply has to understand this from the context?

The other possibility is that I'm not understanding something here. Sorry if this is the case, I'm not fluent in class field theory.