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 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.

A database of number fields

“…The root discriminant of K 1 K 2 then works out to be 2 39/16 3 1/2 7 4/5 ≈ 44.50. The existence of this remarkable compositum contradicts [21,Corollary 12.1] and is the only error we have found in [21].…”

“…As reviewed in the introduction, in [21] we raised the problem of completely understanding the set K[Ω] of all Galois number fields K ⊂ C with grd at most the Serre-Odlyzko constant Ω = 8πe γ ≈ 44.76. As in [21], we focus attention here on the interesting subproblem of identifying the subset K ns [Ω] of K which are nonsolvable. Our last two sections explain how the database explicitly exhibits a substantial part of K ns [Ω].…”

“…In [21], we listed fields proving |K In [21], we highlighted the fact that the only simple groups involved were the five smallest, A 5 , SL 3 (2), A 6 , SL 2 (8), and PSL 2 (11) and the eighth, A 7 . The new fields add SL 2 (16), G 2 (2) , and A 8 to the list of simple groups involved.…”

“…These groups are tenth, twelfth, and tied for nineteenth in the list of all non-abelian simple groups in increasing order of size. (7), Ω], twenty are listed in [21] and three are new. The polynomial for the SL 2 (16) field was found by Bosman [8], starting from a classical modular form of weight two.…”

“…Sections 9 and 10 illustrate progress in the database on a large project initiated in [21]. The project is to completely classify Galois number fields with root discriminant |D| 1/n at most the Serre-Odlyzko constant Ω := 8πe γ ≈ 44.76.…”