Some arithmetical functions in finite fields
Stephen D. Cohen · Glasgow Mathematical Journal · 1970
In this paper, we investigate various “arithmetical” functions associated with the factorisation of polynomials in GF[q, X 1 , …, X k ] , where k ≥ 1 and GF[q] is the finite field of order q . We shall assume throughout that all polynomials discussed are non-zero and have been normalised by selecting one polynomial from each equivalence class with respect to multiplication by non-zero elements of GF[q] . The constant polynomial will be denoted by 1. With this normalisation, GF[q, X 1 , …, X k ] becomes a unique factorisation domain. When k = 1, normalisation is achieved by considering only monic polynomials. By the degree of a polynomial A(X 1 , …, X k ) will be understood the ordered set ( m 1 , …, m k ), where m 1 is the degree of A(X 1 , …, X k ) in X 1 ,( i = 1, …, k ).