Galois number fields with small root discriminant

✍ Scribed by John W. Jones; David P. Roberts

• Book ID: 104024733
• Publisher: Elsevier Science
• Year: 2007
• Tongue: English
• Weight: 318 KB
• Volume: 122
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2006.05.001

✦ Synopsis

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 = A 4 , A 5 , A 6 and S 4 , S 5 , S 6 , finding exactly 59, 78, 5 and 527, 192, 13 fields, respectively. We present other fields with Galois group SL 3 (2), A 7 , S 7 , PGL 2 (7), SL 2 (8), ΣL 2 (8), PGL 2 (9), PΓ L 2 (9), PSL 2 (11), and A 2 5 .2, and root discriminant less than Ω. We conjecture that for all but finitely many groups G, the set K(G, Ω) is empty.