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Let $K$ be a number field and let $S$ be a set of primes of $K$ containing the set of archimedian primes $S_\infty$. Suppose, $S$ has Dirichlet density $\delta(S) = 1$.

Then the claim is that the set of completely splitting primes in the extension $K(\mu_{p^r}) | K$ for every rational prime $p$ and $r \in \mathbb{N}$ arbitrary is contained in $S$ up to a set of Dirichlet density $0$.

My guess is to show, that the Dirichlet density of this set of completely splitting primes is $1$ as well, but I don't know how to apply Chebotarev's density theorem.

Thank you!

BIS HD
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Let $P$ be the set of all primes of $K$. If $S\subseteq P$ has full Dirichlet density $\delta(S)=1$ then we see its complement has density $\delta(P\setminus S)=0$ and hence every subset of $P\setminus S$ has density $0$. Thus if we have any subset $X\subseteq P$, the difference $X\setminus S$ has density $0$, i.e. $X\subseteq S$ up to a set of density $0$.

In your case you can set $X$ to be the set of primes that completely split in all cyclotomic extensions by prime power roots of unity, but these hypotheses on $X$ aren't really needed to see the fact.

anon
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  • Thanks, that was somehow obvious but thanks for the clarification. It helped me a lot! :-) Another question came to my mind: Can one determine the density of the set of completely splitting primes for a finite extension? – BIS HD Mar 13 '14 at 14:23
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    @BISHD Yes we can leverage the Chebotarev density theorem for that purpose - indeed we can stretch it to determine the densities of all possible splitting types in all non-Galois extensions $L/K$ and, more, to determine the densities of all $G_{M/K}$-set isomorphism types (a stronger invariant than splitting type) for Galois extensions $M/K$. How to do this exactly is sufficiently interesting and nontrivial but easy and enjoyable to write about that I recommend asking this as a separate question! – anon Mar 13 '14 at 15:01
  • Thanks for that hint. I asked this question again here: http://math.stackexchange.com/questions/711084/how-to-determine-the-density-of-the-set-of-completely-splitting-primes-for-a-fin – BIS HD Mar 13 '14 at 17:53