Knopfmacher, J. Direct factors of polynomial rings over finite fields. (English) Zbl 0591.12020 J. Comb. Theory, Ser. A 40, 429-434 (1985). An asymptotic formula is derived for the total number of polynomials of degree n in an arbitrary direct factor of the set G of all monic polynomials in one unknown over a finite field with q elements. A direct factor of G is a subset \(B_ 1\) of G such that for some subset \(B_ 2\) of G, every polynomial w in G has a unique factorization of the form \(w=b_ 1b_ 2\), where \(b_ i\in B_ i\). The asymptotic formula is given as \(c_ 1q^ n\), where \(c_ 1\) is a constant depending on \(B_ 1\). Reviewer: J.Liang Cited in 1 Document MSC: 11T06 Polynomials over finite fields 11N45 Asymptotic results on counting functions for algebraic and topological structures Keywords:asymptotic formula; finite field; polynomial PDFBibTeX XMLCite \textit{J. Knopfmacher}, J. Comb. Theory, Ser. A 40, 429--434 (1985; Zbl 0591.12020) Full Text: DOI References: [1] Daboussi, H., On the density of direct factors of the set of positive integers, J. London Math. Soc. (2), 19, 21-24 (1979) · Zbl 0409.10044 [2] Hardy, G. H., Divergent Series (1947), Oxford Univ. Press: Oxford Univ. Press London/New York · Zbl 0897.01044 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.