On the Irreducibility of the Cauchy-Mirimanoff Polynomials

Author

Brian C. Irick, University of Tennessee - KnoxvilleFollow

Date of Award

5-2010

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

Pavlos Tzermias

Committee Members

David Dobbs, Shashikant Mulay, Soren Sorensen

Abstract

The Cauchy-Mirimanoff Polynomials are a class of polynomials that naturally arise in various classical studies of Fermat's Last Theorem. Originally conjectured to be irreducible over 100 years ago, the irreducibility of the Cauchy-Mirimanoff polynomials is still an open conjecture.

This dissertation takes a new approach to the study of the Cauchy-Mirimanoff Polynomials. The reciprocal transform of a self-reciprocal polynomial is defined, and the reciprocal transforms of the Cauchy-Mirimanoff Polynomials are found and studied. Particular attention is given to the Cauchy-Mirimanoff Polynomials with index three times a power of a prime, and it is shown that the Cauchy-Mirimanoff Polynomials of index three times a prime are irreducible.

Recommended Citation

Irick, Brian C., "On the Irreducibility of the Cauchy-Mirimanoff Polynomials. " PhD diss., University of Tennessee, 2010.
https://trace.tennessee.edu/utk_graddiss/707