RUI: Galois Structures in Local Number Fields

University Of Nebraska At Omaha, Omaha NE

Investigators

Abstract

This award supports the investigator's work in algebraic number theory and his collaboration with colleagues. The proposed projects revolve around two central questions in the theory of local number field extensions: What are the invariants that determine the Galois module structure of ideals (and also that of the higher unit groups)? What is the effect of twists by characters of Galois representations on these structures? In addition, this award provides support for undergraduate research. Number Theory is an area of mathematics that begins with the properties of the whole numbers and is currently applied to problems in cryptography and coding theory. Efforts to resolve questions such as Fermat's Last Theorem, have led mathematicians to expand the set of numbers they consider (i.e. beyond the whole numbers, beyond the rational numbers). Fundamental properties of these sets, called algebraic number fields, and the fundamental properties of their localizations, are the focus of the investigators research. The Galois group captures the symmetries of an algebraic number field. Using the Galois group, the investigator proposes to investigate subsets of basic importance ? additive ones that generalize the integers and multiplicative ones that generalize plus/minus one. Beyond supporting the research described above, this award will support undergraduate students who are engaged in number theoretic research with the investigator.