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In the paper of Ihara

https://projecteuclid.org/euclid.jmsj/1230396372,

he proved that the set of primes which split conpletely in

an infinite unramified extension is finite.

In section 13 of the above paper, he explained about an infinite tamely ramified extension cases.

If every ramification index of primes is finite, the answer is yes.

But I am not sure when the ramification index of some prime is infinite.

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  • $\begingroup$ Do you know any case of an infinite extension in which infinitely many primes split completely? $\endgroup$
    – Lubin
    Aug 31, 2016 at 18:03
  • 1
    $\begingroup$ $Z_p$- extension over $Q$ $\endgroup$ Aug 31, 2016 at 23:15

1 Answer 1

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Your claim that the set of primes which split completely in an infinite unramified extension is finite is not true. In fact, Ihara proved the following in his paper:

Let $K$ be a number field and $L/K$ an infinite unramified extension. Denote by $S_f$ the set of prime ideals of $K$ that split completely in $L/K$. Then one has $$ \sum_{\mathfrak{p}\in S_{f}} \dfrac{\log N(\mathfrak{p})}{N(\mathfrak{p})^{f(\mathfrak{p})}-1}<\infty$$

This does not imply that the set $S_f$ is finite. Indeed, Ihara raised the question after Proposition 1: are there $L/K$ for which $S_f$ is infinite? This question was recently answered in the positive by Hajir, Maire, and Ramakrishna in Cutting towers of number fields. See also A note on asymptotically good extensions in which infinitely many primes split completely.

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