TY - JOUR
T1 - Galois number fields with small root discriminant
AU - Jones, John
AU - Roberts, David P.
PY - 2007/2
Y1 - 2007/2
N2 - 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 group SL3 (2), A7, S7, PGL2 (7), SL2 (8), Σ L2 (8), PGL2 (9), P Γ L2 (9), PSL2 (11), and A52 . 2, and root discriminant less than Ω. We conjecture that for all but finitely many groups G, the set K (G, Ω) is empty.
AB - 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 group SL3 (2), A7, S7, PGL2 (7), SL2 (8), Σ L2 (8), PGL2 (9), P Γ L2 (9), PSL2 (11), and A52 . 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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U2 - 10.1016/j.jnt.2006.05.001
DO - 10.1016/j.jnt.2006.05.001
M3 - Article
AN - SCOPUS:43049111326
SN - 0022-314X
VL - 122
SP - 379
EP - 407
JO - Journal of Number Theory
JF - Journal of Number Theory
IS - 2
ER -