Combinatorial Techniques in the Galois Theory of p-Extensions

A major open problem in current Galois theory is to characterize those profinite groups which appear as absolute Galois groups of various fields. Obtaining detailed knowledge of the structure of quotients and subgroup filtrations of Galois groups of p-extensions is an important step toward a solution. We illustrate several techniques for counting Galois p-extensions of various fields, including pythagorean fields and local fields. An expression for the number of extensions of a formally real pythagorean field having Galois group the dihedral group of order 8 is developed. We derive a formula for computing the Fp-dimension of an n-th graded piece of the Zassenhaus filtration for various finitely generated pro-p groups, including free pro-p groups, Demushkin groups and their free pro-p products. Several examples are provided to illustrate the importance of these dimensions in characterizing pro-p Galois groups. We also show that knowledge of small quotients of pro-p Galois groups can provide information regarding the form of relations among the group generators

On the cohomology of Galois groups determined by Witt rings

Let F denote a field of characteristic different from two. In this paper we describe the mod 2 cohomology of a Galois group which is determined by the Witt ring WF

Numerical verification of the Cohen-Lenstra-Martinet heuristics and of Greenberg's pp-rationality conjecture

In this paper we make a series of numerical experiments to support Greenberg's $p$-rationality conjecture, we present a family of $p$-rational biquadratic fields and we find new examples of $p$-rational multiquadratic fields. In the case of multiquadratic and multicubic fields we show that the conjecture is a consequence of the Cohen-Lenstra-Martinet heuristic and of the conjecture of Hofmann and Zhang on the $p$-adic regulator, and we bring new numerical data to support the extensions of these conjectures. We compare the known algorithmic tools and propose some improvements

Galois groups of Schubert problems via homotopy computation

Numerical homotopy continuation of solutions to polynomial equations is the foundation for numerical algebraic geometry, whose development has been driven by applications of mathematics. We use numerical homotopy continuation to investigate the problem in pure mathematics of determining Galois groups in the Schubert calculus. For example, we show by direct computation that the Galois group of the Schubert problem of 3-planes in C^8 meeting 15 fixed 5-planes non-trivially is the full symmetric group S_6006.Comment: 17 pages, 4 figures. 3 references adde

The 22-Class Tower of Q(−5460)\mathbb{Q}(\sqrt{-5460})

The seminal papers in the field of root-discriminant bounds are those of Odlyzko and Martinet. Both papers include the question of whether the field $\mathbb{Q}(\sqrt{-5460})$ has finite or infinite $2$-class tower. This is a critical case that will either substantially lower the best known upper bound for lim inf of root-discriminants (if infinite) or else give a counter-example to what is often termed Martinet's conjecture or question (if finite). Using extensive computation and introducing some new techniques, we give strong evidence that the tower is in fact finite, establishing other properties of its Galois group en route

Maximal unramified 3-extensions of imaginary quadratic fields and SL_2(Z_3)

The structure of the Galois group of the maximal unramified p-extension of an imaginary quadratic field is restricted in various ways. In this paper we construct a family of finite 3-groups satisfying these restrictions. We prove several results about this family and characterize them as finite extensions of certain quotients of a Sylow pro-3 subgroup of SL_2(Z_3). We verify that the first group in the family does indeed arise as such a Galois group and provide a small amount of evidence that this may hold for the other members. If this were the case then it would imply that there is no upper bound on the possible lengths of a finite p-class tower.Comment: 7 pages. No figures. LaTe

Number fields unramified away from 2

We consider finite extensions of the rationals which are unramified except for at 2 and infinity. We show there are no such extensions of degrees 9 through 15