Journal of Number Theory
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.
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John W. Jones and David P. Roberts. Galois number fields with small root discriminant. Journal of Number Theory 122 (2007), no. 2, 379-407.