Polynome über endlichen Körpern mit gleichem Wurzelraum. (Polynomials over finite fields with equal root space) (Q579315)

A refinement of the classical formula is obtained which gives the number of irreducible monic polynomials in one indeterminate of degree n with the coefficients in a finite field \({\mathbb{F}}_ q\). The authors express this number as a sum indexed by the classes of ``root spaces'' corresponding to the polynomials. (The root space of a polynomial is defined as the subspace generated by the roots [see the second author, Elem. Math. 40, 57-59 (1985; Zbl 0572.12003)].) They describe an algorithm for computing the number of irreducible monic polynomials with the root space generated by an element of a group algebra. This algorithm is based on a paper of the authors [J. Algebra 103, 141-159 (1986; Zbl 0607.12011)] about the normal basis theorem and on an explicit formula for the number of regular elements from \({\mathbb{F}}_{q^ n}\). The authors also obtain a necessary and sufficient condition for the equality of the root spaces of two polynomials.