Vol. 85, No. 2, 1979

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Arithmetic properties of the idèle discriminant

Donald Eugene Maurer

Vol. 85 (1979), No. 2, 393–401

A theorem of Hecke asserts that the discriminant dK∕F of an extension of algebraic number fields K∕F a square in the absolute class group. In 1932 Herbrand conjectured the following related theorem and was able to prove it for metacyclic extensions: If K∕F is normal, then dK∕F can be written in the form A2(𝜃), 𝜃 F; where (i) 𝜃 1 (mod )B, B is the greatest divisor of 4 which is prime to dK∕F, and (ii) 𝜃 > 0 at each real prime v except when K FFv is a direct sum of copies of the complex fleld and (K : F) 2 (mod 4).

More recently, A. Fröhlich gave a unified treatment of these and related questions using the concept of an idèle discriminant. The purpose of this paper is to present a generalization of these results with some connections with the structure of the Galois group.

Mathematical Subject Classification
Primary: 12A50, 12A50
Secondary: 12A55
Received: 24 June 1977
Revised: 9 March 1979
Published: 1 December 1979
Donald Eugene Maurer