ACC Seminar: Constructing Totally Ramified Extra-Special p-Extensions of Locals Fields and Resulting Galois Module Structure

ABSTRACT

The normal basis theorem states that if L/K is a Galois extension of fields, then L is a free module of rank one over the group algebra K[G] (G = Gal(L/K)). Naturally, number theorists asked, is there such a thing as an integral basis theorem? That is to say, if L/K is a Galois extension of number fields, with ring of integers D_ L and D_K respectively, is D_L free over D_K[G]? In 1932, E. Noether found that, D_L is locally free over D_K[G] precisely when L/K is at most tamely ramified. That served as a springboard into the local case. In 1959, for a Galois extension of local fields L/K, H.W. Leopoldt introduced the associated order of D_L given by A_(L/K) = {σ ∈ K[G] : σD_K ⊆D_K}. It is well known that A_(L/K) is the only D_K-order of K[G] which D_L can be free over; however, D_L need not be free over its associated order.

Over the past decade, the theory of Galois scaffolds has been developed. It has been discovered that if L/K is a totally ramified Galois extension of local fields which possesses a “precise enough” scaffold, then there are necessary and sufficient conditions for D_L to be free over A_(L/K) which can be stated in terms of the ramification numbers for L/K. Given a Galois extension of local fields L/K, a Galois scaffold for L/K, in essence, is a K-basis for the group ring K[G] (G = Gal(L/K)) whose effect on the valuation of elements of L is easy to determine.

An extra-special p-group is a group G of order p^(2n+1) such that G/Z(G) elementary abelian of order p^(2n). In this talk we will use Artin-Schreier polynomials to construct totally ramified extra-special p-extensions which possess a Galois Scaffold and use the theory of Byott, Childs, and Elder to study the Galois module structure of these extensions. This is joint work with Kevin Keating (University of Florida).

BIOGRAPHY

Paul Schwartz received his PhD from the University of Florida in 2022 under the direction of Kevin Keating. He currently serves as a Lecturer of Mathematics at Stevens Institute of Technology.

Attendance: This is a technical talk open to all.
A campus map is available at https://tour.stevens.edu.
Additional information is available at https://web.stevens.edu/algebraic/.