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is there any conditions over a number field K for an unramified quadratic extension of K to admit an embedding into an unramified cyclic extension of degree 4 of K?

ayoub-chess
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    Does classfield theory answer this adequately for you? – paul garrett Sep 24 '22 at 00:32
  • im not sure. The only thing that i know by class field theory is that if 4-rank of Ideal class group >or egal to 1 then K admit an unramified cyclic extension of degree 4 . – ayoub-chess Sep 24 '22 at 00:43
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    Class field theory says that the Artin map is an isomorphism between the ideal class group and the Galois group of the maximal abelian unramified extension. So questions about Galois group of abelian unramified extensions become questions about the ideal class group. – reuns Sep 24 '22 at 08:04

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