Given an integer n and a group G we estimate the probability that a random, monic, degree n, p-adic polynomial has Galois group isomorphic to G. A comparison is made with classic results concerning Galois groups of random, monic, degree n polynomials with integer coefficients. We conclude by analyzing the probability that a random, monic, degree n polynomial in Z[x] has these properties: its Ga...

We introduce group corings, and study functors between categories of comodules over group corings, and the relationship to graded modules over graded rings. Galois group corings are defined, and a Structure Theorem for the G-comodules over a Galois group coring is given. We study (graded) Morita contexts associated to a group coring. Our theory is applied to group corings associated to a comodu...

Let A and B be R-algebras with automorphism groups G and H respectively. Denote the order of G by n and the order of H by m for some integers n and m. Assume n and m are invertible in R. Then, A⊗R B is a Galois R-algebra with Galois group G × H if and only if A and B are Galois R-algebras with Galois groups G and H respectively. Thus an equivalent condition for a central Galois algebra in terms...

In this paper, we give new constructions of disjoint difference families from Galois rings. The constructions are based on choosing cosets of the unit group of a subring in the Galois ring GR(p2, p2s). Two infinite families of disjoint difference families are obtained from the Galois rings GR(p2, p4n) and GR(22, 22s).

We show that the absolute Galois group of a countable Hilbertian P(seudo)A(lgebraically)C(losed) field of characteristic 0 is a free profinite group of countably infinite rank (Theorem A). As a consequence, G(Q̄/Q) is the extension of groups with a fairly simple structure (e.g., ∏∞ n=2 Sn) by a countably free group. In addition, we characterize those PAC fields over which every finite group is a...