Swan Modules and Hilbert–Speiser Number Fields
✍ Scribed by Cornelius Greither; Daniel R Replogle; Karl Rubin; Anupam Srivastav
- Publisher
- Elsevier Science
- Year
- 1999
- Tongue
- English
- Weight
- 114 KB
- Volume
- 79
- Category
- Article
- ISSN
- 0022-314X
No coin nor oath required. For personal study only.
✦ Synopsis
A number field K is called a Hilbert Speiser field if for each tamely ramified finite abelian extension NÂK the ring of algebraic integers of N, O N , has a normal integral basis over O K , the ring of algebraic integers of K. The classical Hilbert Speiser theorem proves that the field of rational numbers Q is such a field. It is well known that the class number of a Hilbert Speiser field must equal 1. We consider tame elementary abelian extensions of a number field K and Swan modules to obtain additional necessary conditions for K to be a Hilbert Speiser field. We show that among all number fields, the field Q is the only Hilbert Speiser field.