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We show that the S-Euclidean minimum of an ideal class is a rational number, generalizing a result of Cerri. In the proof, we actually obtain a slight refinement of this and give some corollaries which explain the relationship of our results with Lenstra's notion of a norm-Euclidean ideal class and the conjecture of Barnes and Swinnerton-Dyer on quadratic forms. In particular, we resolve a conjecture of Lenstra except when the S-units have rank one. The proof is self-contained but uses ideas from...

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 ${\zeta }_K\left(s\right)$. If $$38{(\ell -1)}^2{(logf)}^{6}loglogf\<f,$$ then $K$ is not norm-Euclidean.