A297063 - OEIS

%I #12 Dec 26 2017 09:16:30

%S 7,9,13,19,31,37,43,61,67,91,103,109,127,157

%N Square roots of discriminants of Galois cubic number fields possessing a norm-Euclidean ideal class.

%C This generalizes A185720, because the unit ideal of a norm-Euclidean number field is a norm-Euclidean ideal. In other words, this sequence consists of the elements of A185720 and 91.

%C There are two Galois cubic number fields with discriminant 91^2; each one possesses a nontrivial norm-Euclidean ideal class.

%C Shigeki Egami showed that there are only finitely many terms in this sequence.

%C Computations by Clark R. Lyons and Kelly Emmrich have shown that this sequence is complete up to 10^6.

%H Shigeki Egami, <a href="https://projecteuclid.org/euclid.tjm/1270153001">On Finiteness of the Numbers of Euclidean Fields in Some Classes of Number Fields</a>, Tokyo J. of Math. Volume 07, Number 1 (1984), pp. 183-198.

%H H. W. Lenstra, Jr., <a href="https://www.math.leidenuniv.nl/~hwl/PUBLICATIONS/1979c/art.pdf">Euclidean ideal classes</a>, Soc. Math. France Astérisque, 1979, pp. 121-131.

%H Kevin J. McGown, <a href="https://arxiv.org/abs/1102.2043">Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis</a>, eprint arXiv:1102.2043, Feb 2011.

%H Kelly Emmrich and Clark Lyons, <a href="https://wcnt.files.wordpress.com/2017/12/wcnt2017-kellyclark.pdf">Norm-Euclidean Ideals in Galois Cubic Fields</a>, Slides, West Coast Number Theory, Dec 18 2017.

%Y Cf. A185720.

%K nonn,fini

%O 1,1

%A _Robert C. Lyons_, Dec 24 2017