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Unramified quaternion extensions of quadratic number fields

Franz Lemmermeyer — 1997

Journal de théorie des nombres de Bordeaux

Classical results of Rédei, Reichardt and Scholz show that unramified cyclic quartic extensions of quadratic number fields k correspond to certain factorizations of its discriminant disc k . In this paper we extend their results to unramified quaternion extensions of k which are normal over , and show how to construct them explicitly.

Binomial squares in pure cubic number fields

Franz Lemmermeyer — 2012

Journal de Théorie des Nombres de Bordeaux

Let K = ( ω ) , with ω 3 = m a positive integer, be a pure cubic number field. We show that the elements α K × whose squares have the form a - ω for rational numbers a form a group isomorphic to the group of rational points on the elliptic curve E m : y 2 = x 3 - m . This result will allow us to construct unramified quadratic extensions of pure cubic number fields K .

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