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][1], [Colmez][2] $\S1.2$) 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.

[1]:https://books.google.cl/books?id=4Vm4Lq1WSHgC&pg=PA455&lpg=PA455&dq=%22on+fixe+un+ensemble+fini+S+de+places%22&source=bl&ots=CWWo25xZsB&sig=ACfU3U2h1eiJgt1s-gO-U1AUiqXPOQSvMQ&hl=es-419&sa=X&ved=2ahUKEwjKpKWnhN_hAhVDHbkGHV1aDmYQ6AEwAXoECAgQAQ#v=onepage&q=%22on%20fixe%20un%20ensemble%20fini%20S%20de%20places%22&f=false

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