arXiv subjects
Publications and source records attributed to Andrew Schultz.
Hilbert's Theorem 90 is a classical result in the theory of cyclic extensions. The quadratic case of Hilbert 90, however, generalizes in noncyclic directions as well. Informed by a poem of Richard Wilbur, the article explores several generalizations, discerning connections among multiplicative groups of fields, values of binary quadratic forms, a bit of module theory over group rings, and even Galois cohomology.
We establish automatic realizations of Galois groups among groups M\rtimes G, where G is a cyclic group of order p^n for a prime p and M is a quotient of the group ring Fp[G].
Assuming the Bloch-Kato Conjecture, we determine precise conditions under which Hilbert 90 is valid for Milnor k-theory and Galois cohomology. In particular, Hilbert 90 holds for degree n when the cohomological dimension of the Galois group of the maximal p-extension of F is at most n.
For positive integers m and r, one can easily show there exist integers N such that for every map D:{1,2,...,N} -> {1,2,...,r} there exist 2m integers x_1 < ... < x_m < y_1 < ... < y_m which satisfy: (a) D(x_1) = ... = D(x_m), (b) D(y_1) = ... = D(y_m), and (c) 2(x_m-x_1) \leq y_m-x_1. In this paper we investigate the minimal such integer, which we call g(m,r). We compute g(m,2) for m \geq 2; g(m,3) for m \geq 4; and g(m,4) for m \geq 3. Furthermore, we consider g(m,r) for general r. Along with results that bound g(m,r), we compute g(m,r) exactly for the following infinite families of r: {f_{2n+3}}, {2f_{2n+3}}, {18f_{2n}-7f_{2n-2}}, and {23f_{2n}-9f_{2n-2}}, where here f_i is the ith Fibonacci number defined by f_0 = 0 and f_1=1.
In the mid-1960s Borevic and Faddeev initiated the study of the Galois module structure of groups of pth-power classes of cyclic extensions K/F of pth-power degree. They determined the structure of these modules in the case when F is a local field. In this paper we determine these Galois modules for all base fields F.