The Tambara Structure of the Trace Ideal for Cyclic Extensions

TL;DR

This paper investigates the Tambara functor structure of trace ideals in cyclic Galois extensions, explicitly determining generators and properties, with applications to finite fields and quadratic forms.

Contribution

It explicitly characterizes the trace ideal generators for cyclic extensions and shows the absolute trace ideal is strongly principal in the Burnside Tambara functor.

Findings

Explicit generators for trace ideals in cyclic extensions

Absolute trace ideal is strongly principal in the Burnside Tambara functor

Calculated trace ideals for finite field extensions

Abstract

This paper explores the Tambara functor structure of the trace ideal of a Galois extension. In the case of a (pro-)cyclic extension, we are able to explicitly determine the generators of the ideal. Furthermore, we show that the absolute trace ideal of a cyclic group is strongly principal when viewed as an ideal of the Burnside Tambara Functor. Applying our results, we calculate the trace ideal for extensions of finite fields. The appendix determines a formula for the norm of a quadratic form over an arbitrary finite extension of a finite field.