Number Fields Ramified at One Prime

For G a finite group and p a prime, a G-p field is a Galois number field K with Gal(K/Q) ∼= G and disc(K) = ±p for some a. We study the existence of G-p fields for fixed G and varying p. For G a finite group and p a prime, we define a G-p field to be a Galois number field K ⊂ C satisfying Gal(K/Q) ∼= G and disc(K) = ±p for some a. Let KG,p denote the finite, and often empty, set of G-p fields. The sets KG,p have been studied mainly from the point of view of fixing p and varying G; see [Har94], for example. We take the opposite point of view, as we fix G and let p vary. Given a finite group G, we let PG be the sequence of primes where each prime p is listed |KG,p| times. We determine, for various groups G, the first few primes in PG and their corresponding fields. Only the primes p dividing |G| can be wildly ramified in a G-p field, and so the sequences PG which are infinite are dominated by tamely ramified fields. In Sections 1, 2, and 3, we consider the cases when G is solvable with length 1, 2, and ≥ 3 respectively, using mainly class field theory. Section 4 deals with the much more difficult case of non-solvable groups, with results obtained by complete computer searches for certain polynomials in degrees 5, 6, and 7. In Section 5, we consider a remarkable PGL2(7)-53 field given by an octic polynomial from the literature. We show that the generalized Riemann hypothesis implies that in fact PPGL2(7) begins with 53. Sections 6 and 7 construct fields for the first primes in PG for more groups G by considering extensions of fields previously found. Finally in Section 8, we conjecture that PG always has a density, and this density is positive if and only if G is cyclic. As a matter of notation, we present G-p fields as splitting fields of polynomials f(x) ∈ Z[x], with f(x) chosen to have minimal degree. When KG,p has exactly one element, we denote this element by KG,p. To avoid a proliferation of subscripts, we impose the convention that m represents a cyclic group of order m. Finally, for odd primes p let p̂ = (−1)(p−1)/4p, so that K2,p is Q( √ p̂). One reason that number fields ramified only at one prime are interesting is that general considerations simplify in this context. For example, the formalism of quadratic lifting as in Section 7 becomes near-trivial. A more specific reason is that algebraic automorphic forms ramified at no primes give rise to number fields ramified at one prime via associated p-adic Galois representations. For example, the fields KS3,23, KS3,31, KS̃4,59 and SL2 (11),11 here all arise in this way in the context of classical modular forms of level one [SD73]. We expect that some of the other fields presented in this paper will likewise arise in similar studies of automorphic forms on larger groups. Most of the computations carried out for this paper made use of pari/gp [PAR06], in both library and command line modes.