A185720 Square roots of discriminants of norm-Euclidean Galois cubic fields.

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

Offset

1,1

Comments

Heilbronn shows that this sequence is finite. McGown 2010 strengthens that result, showing that the largest term is less than 10^70.

Following Godwin & Smith, Lemmermeyer showed that there are no further terms below 500,000.

Theorem 1.1 of McGown 2011: 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.

Links

Table of n, a(n) for n=1..13.

H. J. Godwin and J. R. Smith, On the Euclidean nature of four cyclic cubic fields, Math. Comp. 60:201 (1993), pp. 421-423.

H. Heilbronn, On Euclid's algorithm in cyclic fields, Canadian J. Math. 3 (1951), pp. 257-268.

Franz Lemmermeyer, The Euclidean algorithm in algebraic number fields, 2004.

Kevin J. McGown, Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis, arXiv:1102.2043 [math.NT], 2011.

Kevin J. McGown, Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis, Journal de Théorie des Nombres de Bordeaux, 24 (2012), 425-445.

Kevin J. McGown, Norm-Euclidean Galois fields, arXiv:1011.4501 [math.NT], 2010-2011.

Example

a(10)^2 = 24649 = 157^2.

Crossrefs

Cf. A048981 Squarefree values of n for which the quadratic field Q[sqrt(n)] is Euclidean.

Cf. A297063.

Sequence in context: A027692 A343001 A297063 * A032487 A332103 A160777

Adjacent sequences: A185717 A185718 A185719 * A185721 A185722 A185723

Keyword

nonn,fini

Author

Jonathan Vos Post and Charles R Greathouse IV, Feb 10 2011

Extensions

31, 37, and 43 from Robert C. Lyons, Dec 25 2017

Status

approved