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Norm-Euclidean Galois fields

  • Author(s): McGown, Kevin Joseph;
  • et al.

In this work, we study norm-Euclidean Galois number fields. In the quadratic setting, it is known that there are finitely many and they have been classified. In 1951, Heilbronn showed that for each odd prime l, there are finitely many norm-Euclidean Galois fields of degree l. Unfortunately, his proof does not provide an upper bound on the discriminant, even in the cubic case. We give, for the first time, an upper bound on the discriminant for this class of fields. Namely, for each odd prime l we give an upper bound on the discriminant of norm-Euclidean Galois fields of degree l. In Chapter 3, we derive various inequalities which guarantee the failure of the norm- Euclidean property. Our inequalities involve the existence of small integers satisfying certain splitting and congruence conditions; this reduces the problem to the study of character non-residues. This also leads to an algorithm for tabulating a list of candidate norm- Euclidean Galois fields (of prime degree l up to a given discriminant. We have implemented this algorithm and give some numerical results when l < 30. The cubic case is especially interesting as Godwin and Smith have classified all norm-Euclidean Galois cubic fields with [Delta] < 10⁸. Using an efficient implementation of our algorithm, we extend their classification to include all fields with [Delta] < 10²⁰. In Chapter 4, we turn to the study of character non-residues. In sect. 4.1, we give a new estimate of the second smallest prime non-residue, and in sect. 4.2, we derive an explicit version of a character sum estimate due to Burgess following a method of Iwaniec. In Chapter 5, we combine a result of Norton on the smallest non-residue with our results from Chapter 4 to obtain the aforementioned discriminant bounds. In Chapter 6, we give strengthened versions of all our results assuming the Generalized Riemann Hypothesis. Finally, in Chapter 7, we summarize what our results say in the cubic case and use a combination of theory and computation to give, assuming the GRH, a complete determination of all norm-Euclidean Galois cubic fields

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