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Let $p$ be a fixed prime.

Question:

  1. For any number field $K$, is there always a finite extension $L$ of $K$ of $p$-power order such that the class number of $L$ is prime to $p$?
  2. Moreover, for any number field $K$ and any finite set $S$ of primes of $K$, is there always a finite extension $L$ of $K$ of $p$-power order such that $ L/K $ is unramified at any prime outside $ S $ the class number of $L$ is prime to $p$?
Improve this question
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  • $\begingroup$ The answer to the question is No. One can show that a number field $K$ can be imbeded into another number field with class number prime to $p$ iff the degree of the maximal unramified $p$-extension (or $p$-class field tower) of $K$ is finite. But there are infinitely many examples of number fields with infinite $p$-class field tower. See Proposition 2 and Corollary 7 on p.231-234 of Algebraic Number Theory by Cassels and Frohlich. $\endgroup$
    – stupid boy
    Dec 6, 2022 at 14:29

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