Assuming the generalized Riemann hypothesis, we prove the following complexity bounds: The order of the Galois group of an arbitrary polynomial f(x) ∈ Z[x] can be computed in P#P. Furthermore, the order can be approximated by a randomized polynomial-time algorithm with access to an NP oracle. For polynomials f with solvable Galois group we show that the order can be computed exactly by a random...

We study the problem of finding the Artin L-functions with the smallest conductor for a given Galois type. We adapt standard analytic techniques to our novel situation of fixed Galois type and obtain much improved lower bounds on the smallest conductor. For small Galois types we use complete tables of number fields to determine the actual smallest conductor.

We prove Serre’s conjecture for the case of Galois representations with Serre’s weight 2 and level 1. We do this by combining the potential modularity results of Taylor and lowering the level for Hilbert modular forms with a Galois descent argument, properties of universal deformation rings, and the non-existence of p-adic Barsotti-Tate conductor 1 Galois representations proved in [Di3].

If the absolute Galois group GK of a field K is a direct product GK = G1 × G2 then one of the factors is prosolvable and either G1 and G2 have coprime order or K is henselian and the direct product decomposition reflects the ramification structure of GK . So, typically, the direct product of two absolute Galois groups is not an absolute Galois group. In contrast, free (profinite) products of ab...

In earlier eighties of XX century A.A.Nechaev has obtained the criterion of full period of a Galois polynomial over primary residue ring Z2n . Also he has obtained necessary conditions of maximal period of the Galois polynomial over Z2n in terms of coefficients of this polynomial. Further A.S.Kuzmin has obtained analogous results for the case of Galois polynomial over primary residue ring of od...

Consider any nilpotent group G of finite odd order. We ask if we can always find a galois extension K of Q such that Gal(K/Q) ∼= G. This is the famous Inverse Galois Problem applied to nilpotent groups of finite odd order. By solving the Group Extension Problem and the Embedding Problem, two problems that are related to the Inverse Galois Problem, we show that such a K always exists. A major re...

Generalising the notion of Galois corings, Galois comodules were introduced as comodules P over an A-coring C for which PA is finitely generated and projective and the evaluation map μC : Hom (P, C) ⊗S P → C is an isomorphism (of corings) where S = End(P ). It was observed that for such comodules the functors HomA(P,−)⊗S P and −⊗A C from the category of right A-modules to the category of right ...

We investigate double transitivity of Galois groups in the classical Schubert calculus on Grassmannians. We show that all Schubert problems on Grassmannians of 2and 3-planes have doubly transitive Galois groups, as do all Schubert problems involving only special Schubert conditions. We use these results to give a new proof that Schubert problems on Grassmannians of 2-planes have Galois groups t...

Arboreal Galois groups sit naturally as subgroups of tree (or graph) automorphism groups, while dynatomic Galois groups are naturally subgroups of certain wreath products. A fundamental problem is to determine general conditions under which these dynamically generated Galois groups have finite index in the natural geometric groups that contain them. This is a dynamical analog of Serre’s theorem...

In 1892, D. Hilbert began what is now called Inverse Galois Theory by showing that for each positive integer m, there exists a polynomial of degree m with rational coefficients and associated Galois group Sm, the symmetric group on m letters, and there exists a polynomial of degree m with rational coefficients and associated Galois group Am, the alternating group on m letters. In the late 1920’...