Galois number fields with small root discriminant

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 = 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 ) , Σ L 2 ( 8 ) , PGL 2 ( 9 ) , P Γ L 2 ( 9 ) , PSL 2 ( 11 ) , and A 5 2 . 2 , and root discriminant less than Ω. We conjecture that for all but finitely many groups G, the set K ( G , Ω ) is empty.