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.*][1] 
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. 

[1]:http://www.numdam.org/item/SB_1998-1999__41__21_0/