$p > 2$ and ramification of archimedean places

Fix a rational prime $p$. I know that for a $p$-extension (ie. a Galois extension of degree a power of $p$) of an algebraic number field $k$, some places can not ramify:

1. complex places cannot ramify.
2. real places cannot ramify if $p\neq 2$.

This is presumably easy to solve but if we assume $p > 2$, archimedean places cannot ramify. Let $S$ be a finite set of places of $k$. Can we assume $S$ to contain those archimedean places and does this alter the maximal $p$-extension that is unramified outside $S$?

Thanks! :-)

• Just to make sure, are you asking whether or not for a $p$-extension if adding the infinite places to any finite set of places $S$ changes what the maximal $p$-extension unramified outside of $S$ is? – Alex Youcis Nov 29 '13 at 2:06
• @AlexYoucis yes, exactly! – BIS HD Nov 29 '13 at 11:21
• @AlexYoucis Do you have an idea or a hint for that? – BIS HD Dec 6 '13 at 16:04
• Yes, for $p$ odd, you can assume that S contains the archimedean primes, and no, this does not alter the max. pro-$p$-extension unramified outside S. – nguyen quang do Feb 14 at 8:00