Dirichlet's theorem for the ring of polynomials over \(\mathrm{GF}(2)\)
type: Item
instance of: scholarly article
title: Dirichlet's theorem for the ring of polynomials over \(\mathrm{GF}(2)\)
zbMATH Open document ID: 0582.12010
author: Douglas Hensley
published in: Pacific Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: Let \(G\) denote the ring \(\mathrm{GF}(2)[x]\) of polynomials \(p\) over the field of integers mod 2. Let \(I(k)=\#\{p: \deg p=k\) and \(p\) is irreducible in \(G\}\). A well known analog of the prime number theorem is that \(I(k)=(1/k) \sum_{d\mid k}\mu(d) 2^{k/d}\approx (1/k) 2^k.\) This is extended to an analog of Dirichlet's theorem on primes in arithmetic progressions, with special emphasis on the square-free case. The approach is by way of ``\(L\)-functions'', but in this simpler setting series expansions and coefficient estimates take the place of contour integration.
Mathematics Subject Classification ID: 11T06
Mathematics Subject Classification ID: 11N13
Mathematics Subject Classification ID: 11T99
zbMATH DE Number: 3931145
zbMATH Keywords: analog of Dirichlet's theorem
zbMATH Keywords: primes in arithmetic progressions
zbMATH Keywords: L-functions
zbMATH Keywords: number of irreducible polynomials
MaRDI profile type: Publication
full work available at URL: pjm.1986.123.93
OpenAlex ID: W1989167556
DOI: 10.2140/PJM.1986.123.93
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description: scientific article; zbMATH DE number 3931145