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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! :-)

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  • $\begingroup$ 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? $\endgroup$ – Alex Youcis Nov 29 '13 at 2:06
  • $\begingroup$ @AlexYoucis yes, exactly! $\endgroup$ – BIS HD Nov 29 '13 at 11:21
  • $\begingroup$ @AlexYoucis Do you have an idea or a hint for that? $\endgroup$ – BIS HD Dec 6 '13 at 16:04
  • $\begingroup$ 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. $\endgroup$ – nguyen quang do Feb 14 at 8:00

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