J Austral Math Soc Ser A 51 pp171--186, 1991.

Rigidity of Decomposition Laws and Number Fields

Norbert Klingen

Abstract

We speak of rigidity, if patial information about the prime decomposition in an extension of number fields K/k determines the decomposition law completely (and hence the zeta function z_K), or even fixes the field K itself. Several concepts of rigidity, depending on the degree of information we start from, are introduced and studied. The strongest concept (absolute rigidity) was only known to hold for the ground field and all quadratic extensions. Here a complete list of all Galois quartic extensions which are absolutely rigid is given. For the weaker concept of rigidity, all rigid situations among the fields of degree up to 8 are determined.

1980 AMS Subject Classification (1985 Revision): 11R32, 11R27, 20B25

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Authors

Norbert Klingen

Mathematiches Institut, Universität zu Köln, Weyertal 86-90, D 5000 Köln 41, Germany.