Direct factors of polynomial rings over finite fields

AbstractLet Gq denote the multiplicative semigroup of all monic polynomials in one indeterminate over a finite field Fq with q elements. By a direct factor of Gq is understood a subset B1 of Gq such that, for some subset B2 of Gq, every polynomial w ∈ Gq has a unique factorization in the form w = b1b2 for bi ∈ Bi. An asymptotic formula B1#(n) ∼ c1qn as n → ∞ is derived for the total number B1#(n) of polynomials of degree n in an arbitrary direct factor B1 of Gq, c1 a constant depending on B1