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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.

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    $\begingroup$ When $S$ is a finite set of places that does not mean it must contain $S_\infty$. Yet often in practice it does contain $S_\infty$ and this restriction should be explicitly stated when it occurs. Are you sure you're not overlooking some convention earlier in a paper that would imply the sets $S$ in the paper will always contain $S_\infty$? Please give some examples of papers where you think the author (say Serre) is forgetting to say $S \supset S_\infty$. $\endgroup$
    – KConrad
    Commented Apr 20, 2019 at 6:41
  • $\begingroup$ @KConrad I'll put the links in my question. $\endgroup$
    – efs
    Commented Apr 20, 2019 at 15:54
  • $\begingroup$ Please put actual article references in the post, not just links. When I click on the first link, I am sent to Google Books and it tells me in Spanish that I've reached my page limit. $\endgroup$
    – KConrad
    Commented Apr 20, 2019 at 18:27
  • $\begingroup$ @KConrad Sorry, I hope now is ok. $\endgroup$
    – efs
    Commented Apr 20, 2019 at 18:40
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    $\begingroup$ @EFinat-S You're right, in Serre 2.1 it seems the assumption $S \supset S_\infty$ is missing. I am not familiar enough with the literature to tell whether some authors leave ramification at infinity undefined, which would explain this. $\endgroup$
    – François Brunault
    Commented Apr 23, 2019 at 4:50

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