Table of Contents
Journal of Discrete Mathematics
Volume 2014 (2014), Article ID 263179, 4 pages
http://dx.doi.org/10.1155/2014/263179
Research Article

Counting Irreducible Polynomials of Degree over and Generating Goppa Codes Using the Lattice of Subfields of

1Malawi Institute of Technology, Malawi University of Science and Technology, P.O. Box 5196, Limbe, Malawi
2Department of Mathematics, Mzuzu University, Private Bag 201, Luwinga, Mzuzu, Malawi

Received 28 May 2014; Accepted 8 September 2014; Published 18 September 2014

Academic Editor: Jean-Pierre Gazeau

Copyright © 2014 Kondwani Magamba and John A. Ryan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The problem of finding the number of irreducible monic polynomials of degree over is considered in this paper. By considering the fact that an irreducible polynomial of degree over has a root in a subfield of if and only if , we show that Gauss’s formula for the number of monic irreducible polynomials can be derived by merely considering the lattice of subfields of . We also use the lattice of subfields of to determine if it is possible to generate a Goppa code using an element lying in a proper subfield of .