A297063 Square roots of discriminants of Galois cubic number fields possessing a norm-Euclidean ideal class.

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

Offset

1,1

Comments

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.

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

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

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

Links

Table of n, a(n) for n=1..14.

Shigeki Egami, On Finiteness of the Numbers of Euclidean Fields in Some Classes of Number Fields, Tokyo J. of Math. Volume 07, Number 1 (1984), pp. 183-198.

H. W. Lenstra, Jr., Euclidean ideal classes, Soc. Math. France Astérisque, 1979, pp. 121-131.

Kevin J. McGown, Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis, eprint arXiv:1102.2043, Feb 2011.

Kelly Emmrich and Clark Lyons, Norm-Euclidean Ideals in Galois Cubic Fields, Slides, West Coast Number Theory, Dec 18 2017.

Crossrefs

Cf. A185720.

Sequence in context: A125866 A027692 A343001 * A185720 A032487 A332103

Adjacent sequences: A297060 A297061 A297062 * A297064 A297065 A297066

Keyword

nonn,fini

Author

Robert C. Lyons, Dec 24 2017

Status

approved