Let K/Q be a galois extension, p an odd prime and L/K a Z_p extension, different from the cyclotomic one. Let H/K be finite abelian and linearly disjoint from L.

a) Are there infinitely many primes of Q that are totally split in L/K? b) If yes, is there some limit of the Tchebotarew density theorem one can apply to L?

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    $\begingroup$ The subject line ("CD - continuous development") could really be improved. $\endgroup$ – KConrad Nov 7 '14 at 21:13
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    $\begingroup$ How does $H$ come into the question? $\endgroup$ – Gerry Myerson Nov 7 '14 at 22:22

It may be the case that no prime $v$ of $K$ splits completely in $L$. Indeed, that would be the "typical" outcome.

If there exists at least one non-cyclotomic $\mathbf{Z}_p$-extension of $K$, there exists a $\mathbf{Z}^2_p$-extension $M/K$ which contains the cyclotomic extension. Let $\Gamma = \mathrm{Gal}(M/K)$. Suppose that $v \nmid p$. Because $v$ does not split completely in the cyclotomic extension, the decomposition group $D_v \subset \Gamma$ is isomorphic to $\mathbf{Z}_p$ (It is cyclic and non-trivial). A choice of $\mathbf{Z}_p$ extension corresponds to a choice of saturated subgroup

$$\mathbf{Z}_p \simeq H \subset \Gamma,$$

where $\mathrm{Gal}(L/K) = \Gamma/H$.

The prime $v$ splits completely in $L/K$ if and only if $H$ contains the subgroup $D_v \simeq \mathbf{Z}_p $ of $\Gamma$. Now there are uncoutably many choices of subgroup $H$ (and hence choices of $L$) and only finitely many contain any fixed subgroup $D_v$, and so only countably many contain a $D_v$ as $v$ ranges over all primes in $K$. Hence, for a "generic" $\mathbf{Z}_p$ extension, no single prime $v$ will split completely. (It's easy to incorporate into this argument the primes $v|p$ as well.)

I'm not sure if this answers your question or not

  • $\begingroup$ So, your $H$ is not the $H$ in the original question. $\endgroup$ – Gerry Myerson Nov 9 '14 at 5:05
  • $\begingroup$ To user61509 - So a simple cardinality argument shows there are many $\mathbb{Z}_p$-extensions in which no prime is totally split - actually uncountably many, while the other ones are countable. The question was referring to the cases in which there \textit{are} tottally split primes, and in that case I was wondering what "proportion" of them would split on in the extension $H/L$. The question includes of course the need of a definition of "proportion", since we do not know a priori if there are finitely or infinitely many primes that are totally split. $\endgroup$ – Preda Dec 25 '16 at 12:06

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