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The number of generators of the integers of a number field

Published online by Cambridge University Press:  26 February 2010

P. A. B. Pleasants
P. A. B. Pleasants
Affiliation:
University College, Cardiff.
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Extract

This paper deals with the problem (raised by J. Browkin) of how many ring generators are needed for the ring of integers of a given algebraic number field. I show that the number of generators needed for the integers of a field of degree n is less than (logn/log2) + 1, and that if 2 splits completely in the field the number of generators needed is in fact the largest integer less than (logn/log2) + 1. These results follow from a computable formula (that depends only on how the small primes factorize in the field) for the number of generators of the ring of integers of a given field. This formula has the single drawback that when it yields “one” two generators may be needed, and I show that there are fields of arbitrarily high degree for which this happens.

Type
Research Article
Information
Mathematika , Volume 21 , Issue 2 , December 1974 , pp. 160 - 167
Copyright
Copyright © University College London 1974

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