Paul SchwartzAffiliation: Stevens Institute of Technology Title Of Talk: Constructing Totally Ramified Extra-Special $p$-Extensions of Locals Fields and Resulting Galois Module Structure.
URLS: 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 $K[G]$ ($G=\text{Gal}(L/K))$. 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 $\mathfrak{O}_L $ and $\mathfrak{O}_K$ respectively, is $\mathfrak{O}_L$ free over $\mathfrak{O}_K[G]$? In 1932, E. Noether found that, $\mathfrak{O}_L$ is locally free over $\mathfrak{O}_K[G]$ precisely when $L/K$ is at most tamely ramified. That served as a spring board into the local case. In 1959, for a Galois extension of local fields $L/K$, H.W. Leopoldt introduced the associated order of $\mathfrak{O}_L$ given by $\mathfrak{A}_{L/K}=\{\sigma\in K[G]:\sigma\mathfrak{O}_K\subseteq \mathfrak{O}_K \}$. It is well know that $\mathfrak{A}_{L/K}$ is the only $\mathfrak{O}_K$-order of $K[G]$ which $\mathfrak{O}_L$ can be free over; however, $\mathfrak{O}_L$ 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 $\mathfrak{O}_L$ to be free over $\mathfrak{A}_{L/K}$ 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 $K[G]$ ($G= \text{Gal}(L/K)$) 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 $G/Z(G) \cong \mathbb{Z}^{2n}$. 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).
WARNING: This page contains MATH-JAX
Last update made Tue Mar 10 21:28:05 CDT 2026.
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