Wild Ramification Bounds and Simple Group Galois Extensions Ramified Only at 2

We consider finite Galois extensions of Qp and deduce bounds on the discriminant of such an extension based on the structure of its Galois group. We then apply these bounds to show that there are no Galois extensions of Q, unramified outside of {2,∞}, whose Galois group is one of various finite simple groups. The set of excluded finite simple groups includes several infinite families. Understanding the Galois extensions of Q in terms of their Galois groups and sets of ramifying primes is one of the central goals of algebraic number theory. Here, we consider the problem from the perspective of severely limiting the set of ramifying primes and trying to understand what Galois extensions and Galois groups can then occur. Let K2 be the set of finite Galois extensions of Q in C which are unramified outside of the set {2,∞}, and let G2 := {Gal(K/Q) | K ∈ K2}. The sets K2 and G2 have been studied in several papers [Tat94, Har94, Bru01, Les, Mar63, Moo07, Jon]. One can restrict ramification even further and consider K 2 , the set of totally real fields in K2 and G 2 = {Gal(K/Q) | K ∈ K + 2 }. Relatively few examples are known of groups in G2, and fewer in G 2 . The case of 2-groups is fully understood by [Mar63]. The smallest non-2-group in G2 is C17 : C16 [Har94], and recently, Dembélé [Dem09] has shown that G2 contains the non-solvable group, SL2(2).C8. We consider non-abelian finite simple groups and prove that in many cases, the group in question is either not in G2 or in G 2 . We focus on simple groups for two reasons. Much of the work in the area of studying extensions with restricted ramification makes use of class field theory, and non-abelian simple groups force us to develop and use other techniques. More importantly, any extension K ∈ K2 can be viewed as a tower of Galois extensions where each step has Galois group being a simple group. The first step then comes from a simple group in G2. In all known examples of K ∈ K2, this simple group is C2, the cyclic group of order 2. So, it is natural to ask which, if any, non-abelian simple groups are in G2. We will prove that various groups are not in G2 or G 2 by discriminant bound arguments. We will make use of known bounds for root discriminants of number fields as developed by of Odlyzko, Serre, et.al.[Odl90, Ser86], and known techniques for computing similar bounds. Our principle objective then is to derive upper bounds on the contribution of a prime p to the discriminant of a Galois field based on the Galois group of the extension. In Section 1, we review some background on higher ramification groups and the slope content of an extension. Section 2.1 defines the composita indices of a finite p-group, and shows how to use them to derive discriminant bounds. The work most 2010 Mathematics Subject Classification. Primary 11R21, 11S15.