A proof that Z[square root 14] is a Euclidean domain /

• Main Author: Harper, Malcolm, 1958-
• Other Authors: Murty, M. Ram (advisor)
• Format: Others
• Language: en
• Published: McGill University 2000
• Subjects: Mathematics.
• Online Access: http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=36811

This thesis provides the first unconditional proof that the ring Z&sqbl0;14&sqbr0; is a Euclidean domain. The proof is generalized to other real quadratic fields and to cyclotomic extensions of Q . It is proved that when K is a real quadratic field (modulo the existence of two special primes of K) and when K is a cyclotomic extension of Q that OK satisfies: OK is a Euclidean domain if and only if it is a principal ideal domain. === The final chapter discusses the good possibility of finding similar results when K is an arbitrary Galois extension of Q . === The proof is a modification of the proof of a theorem of Clark and Murty giving a similar result when K is a totally real extension of degree at least three. The main changes are a new Motzkin-type lemma and the addition of the large sieve to the argument. These changes allow application of a powerful theorem due to Bombieri, Friedlander and Iwaniec in order to obtain the result in the real quadratic case. The modification also allows the completion of the classification of cyclotomic extensions in terms of the Euclidean property and is expected to yield an advance in the general case.