For a number field , let denote its Hilbert -class field, and put . We will determine all imaginary quadratic number fields such that is abelian or metacyclic, and we will give in terms of generators and relations.
Classical results of Rédei, Reichardt and Scholz show that unramified cyclic quartic extensions of quadratic number fields correspond to certain factorizations of its discriminant disc . In this paper we extend their results to unramified quaternion extensions of which are normal over , and show how to construct them explicitly.
In this note we give a new proof of the theorem of Kronecker-Weber based on Kummer theory and Stickelberger’s theorem.
Let , with a positive integer, be a pure cubic number field. We show that the elements whose squares have the form for rational numbers form a group isomorphic to the group of rational points on the elliptic curve . This result will allow us to construct unramified quadratic extensions of pure cubic number fields .
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