Counting irreducible polynomials of degree \(r\) over \(\mathbb F_{q^n}\) and generating Goppa codes using the lattice of subfields of \(\mathbb F_{q^{nr}}\) (Q470971)

Summary: The problem of finding the number of irreducible monic polynomials of degree \(r\) over \(\mathbb F_{q^{n}}\) is considered in this paper. By considering the fact that an irreducible polynomial of degree \(r\) over \(\mathbb F_{q^{n}}\) has a root in a subfield \(\mathbb F_{q^{s}}\) of \(\mathbb F_{q^{nr}}\) if and only if \((nr/s,r) = 1\), we show that Gauss's formula for the number of monic irreducible polynomials can be derived by merely considering the lattice of subfields of \(\mathbb F_{q^{nr}}\). We also use the lattice of subfields of \(\mathbb F_{q^{nr}}\) to determine if it is possible to generate a Goppa code using an element lying in a proper subfield of \(\mathbb F_{q^{nr}}\).