Document Type

Article

Publication Date

2-2007

Publication Title

Journal of Number Theory

Abstract

We pose the problem of identifying the set K(G,Ω) of Galois number fields with given Galois group G and root discriminant less than the Serre constant Ω ≈ 44.7632. We definitively treat the cases G = A4. A5, A6, and S4, S5, S6, finding exactly 59, 78, 5 and 527, 192, 13 fields, respectively. We present other fields with Galois groups SL3(2), A7, S7, PGL2(7), SL2(8), ΣL2(8), PGL2(9), PSL2(11), and A25.2, and root discriminant less than Ω. We conjecture that for all but finitely many groups G, the set K(G,Ω) is empty.

Volume

122

Issue

2

First Page

379

Last Page

407

DOI

https://doi.org/10.1016/j.jnt.2006.05.001

ISSN

0022-314X

Comments

This is a pre-publication version of an article published in Journal of Number Theory. The final authenticated version is available online at: https://doi.org/10.1016/j.jnt.2006.05.001

Rights

© 2006 Elsevier Inc. All rights reserved.

Included in

Number Theory Commons

Share

COinS