Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis

TL;DR

Under the assumption of the Generalized Riemann Hypothesis, this paper classifies norm-Euclidean Galois cubic fields and provides a criterion to exclude certain Galois number fields of prime degree from being norm-Euclidean.

Contribution

It establishes a complete classification of norm-Euclidean Galois cubic fields assuming GRH and introduces a general criterion for non-norm-Euclidean Galois fields of prime degree.

Findings

Norm-Euclidean Galois cubic fields are exactly those with specific discriminants.

A new criterion under GRH to determine non-norm-Euclidean Galois fields of prime degree.

Conditional classification based on discriminants and field conductors.

Abstract

Assuming the Generalized Riemann Hypothesis (GRH), we show that the norm-Euclidean Galois cubic fields are exactly those with discriminant $\Delta=7^2,9^2,13^2,19^2,31^2,37^2,43^2,61^2,67^2,103^2,109^2,127^2,157^2$. A large part of the proof is in establishing the following more general result: Let $K$ be a Galois number field of odd prime degree $\ell$ and conductor $f$. Assume the GRH for $\zeta_K(s)$. If $38(\ell-1)^2(\log f)^6\log\log f<f$, then $K$ is not norm-Euclidean.