Counting Irreducible Polynomials of Degree r over Fqn and Generating Goppa Codes Using the Lattice of Subfields of Fqnr. (17th September 2014) - Magamba, Kondwani; Ryan, John A.
Abstract : The problem of finding the number of irreducible monic polynomials of degreer overF q n is considered in this paper. By considering the fact that an irreducible polynomial of degreer overF q n has a root in a subfieldF q s ofF q n r if and only if( n r / s, r ) = 1, we show that Gauss's formula for the number of monic irreducible polynomials can be derived by merely considering the lattice of subfields ofF q n r . We also use the lattice of subfields ofF q n r to determine if it is possible to generate a Goppa code using an element lying in a proper subfield ofF q n r .