COUNTING IRREDUCIBLE FACTORS OF POLYNOMIALS OVER A FINITE-FIELD

Let F(q)[X] denote a polynomial ring in an indeterminate X over a finite field F(q). Exact formulae are derived for (i) the number of polynomials of degree n in F(q)[X] with a specified number of irreducible factors of a fixed degree r in F(q)[X]and (ii) the average number of such irreducible factors and corresponding variance for a polynomial of degree n in F(q)[X]. The main emphasis is on the case when multiplicity of factors is counted. These results are then applied to derive the mean and variance for the total number of irreducible factors of polynomials of degree n in F(q)[X].