Maximal 2-Extensions with Restricted Ramification

Abstract

We compute the Galois groups of several 2-extensions of ramified at finitely many odd primes.

A database of number fields

“…In comparison with the nonsolvable cases discussed in the previous two sections, there is much greater regularity in this setting. We exhibit some of the greater regularity and explain how it makes some of the abstract considerations of [9,10] more concrete. Twenty-six of the 50 octic groups have 2-power order.…”

“…The connection with [9,10] is as follows. Let L(p, q) k ⊂ C be the splitting field of all degree-2 k fields with Galois group a 2-group and discriminant − * p * q * .…”

“…The group Gal(L(p, q)/Q) is a pro-2-group generated by the tame ramification elements τ p and τ q . The central question pursued in [9,10] is the distribution of the Gal(L(p, q)/Q) as abstract groups. Table 7.1 corresponds to working at the level of the quotient Gal(L(p, q) 3 /Q).…”

“…Sections 7 and 8 continue to pursue cases with D = − s p a q b , but now for octic groups G of 2-power order. Section 7 treats the cases p > 2 and discusses connections to tame maximal nilpotent extensions as studied in [9,10]. Section 8 treats the case p = 2 and takes a first step towards understanding wild ramification in some of the nilpotent extensions studied in [26].…”

Unramified Subextensions of Ray Class Field Towers

“…After this paper was written, Lemmermeyer informed us that Part (1) of this theorem, with the same proof, can be found in [22]. For further results about the structure of G T for small sets T , see Boston and Perry [4], Boston and Leedham-Green [5], and Koch [20].…”