Unramified Quadratic Extensions of Real Quadratic Fields, Normal Integral Bases, and 2-AdicL-Functions
✍ Scribed by A Srivastav; S Venkataraman
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 320 KB
- Volume
- 67
- Category
- Article
- ISSN
- 0022-314X
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✦ Synopsis
Let K=Q(-m) be a real quadratic number field. In this article, we find a necessary and sufficient condition for K to admit an unramified quadratic extension with a normal integral basis distinct from K(-&1), provided that the prime 2 splits neither in KÂQ nor in Q(-&m)ÂQ, in terms of a congruence satisfied by the value of a 2-adic L-function for K at 1. 1997 Academic Press 1. INTRODUCTION Throughout this note, the ring of integers of a number field F will be denoted by O F . Let NÂK be a finite Galois extension of number fields with group G. The study of the structure of O N as a G-module has developed into a rich and flourishing area of research. If the extension NÂK is at most tamely ramified then M. J. Taylor's well known proof [4] of Fro lich's conjecture shows that the structure of O N as a Z[G] module is dominated by the associated Artin L-functions. However, not much is known for the relative integral Galois module structure of O N (i.e., as a O K [G]-module).