Galois number fields with small root discriminant Original Research Article

We pose the problem of identifying the set image 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 group SL3(2), A7, S7, PGL2(7), SL2(8), ΣL2(8), PGL2(9), PΓL2(9), PSL2(11), and image, and root discriminant less than Ω. We conjecture that for all but finitely many groups G, the set image is empty.