Dirichlet’s theorem for the ring of polynomials over GF(2)

Douglas Austin Hensley

Vol. 123 (1986), No. 1, 93–101

DOI: 10.2140/pjm.1986.123.93

Abstract

Let G denote the ring GF(2)[x] of polynomials g(x) over the field of integers mod 2. Let

It is well known that I(k) = (1∕k)Σ_d|kμ(d)2^k∕d. Here we prove an analog to Dirichlet’s Theorem on primes in arithmetic progressions. For any m ∈ G the p counted in I(k) are uniformly distributed among the congruence classes (b) mod m for which (b,m) = 1. The result is especially sharp when m is square-free.

Mathematical Subject Classification 2000

Primary: 11T06

Secondary: 11N45

Milestones

Received: 28 December 1984

Published: 1 May 1986