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Euclidean Rings of Algebraic Integers |
Canad. J. Math. 56(2004), 71-76
Published:2004-02-01
Printed: Feb 2004
Printed: Feb 2004
- Malcolm Harper
- M. Ram Murty
Abstract
Let $K$ be a finite Galois extension of the field of rational numbers
with unit rank greater than~3. We prove that the ring of integers of
$K$ is a Euclidean domain if and only if it is a principal ideal
domain. This was previously known under the assumption of the
generalized Riemann hypothesis for Dedekind zeta functions. We now
prove this unconditionally.
| MSC Classifications: |
11R04 - Algebraic numbers; rings of algebraic integers 11R27 - Units and factorization 11R32 - Galois theory 11R42 - Zeta functions and $L$-functions of number fields [See also 11M41, 19F27] 11N36 - Applications of sieve methods |