An Eisenstein Criterion for Noncommutative Polynomials

An analogue of the Eisenstein irreducibility criterion is developed for linear differential operators, or, more generally, noncommutative polynomials, and is applied to a few simple examples. Introduction. The question of the irreducibility of an ordinary linear differential operator is of interest in the Picard-Vessiot theory (see Kolchin [1, §22]). Indeed, the operator is irreducible if and only if the Galois group is linearly irreducible. In this note we develop an "Eisenstein criterion" to help answer this question. Because a linear differential operator is a special type of noncommutative polynomial, it is more natural to phrase the result in this more general context. The result, and indeed the proof with only slight changes in notation, is valid for partial differential operators, or more generally, noncommutative polynomials in several variables. Because the hypothesis that the ring be principal is unnatural in this setting, and because the reader may make the necessary changes if he so desires, we confine our attention to the ordinary case. 1. The irreducibility criterion. By a differential ring ^ we shall mean a ring (associative, commutative, with unit) together with two commuting operators a and ô such that o is an automorphism of SM, à is additive, and ô(ab)=oa-ôb+ôa-b for every a, b in 3%. ¿%[ô] will denote the noncommutative ring of differential operators with coefficients in ÛI, so that ¡%{b] is the set of 2"=o M'> with a¿ e SU, n e N, and right multiplication is defined by ô-a=aa-ô+ôa. Evidently if cr=id, then M is an ordinary differential ring with derivation operator b. By a differential ideal of M we mean an ideal which is closed under the two operators a and ô. The smallest differential ideal which contains the set 2¡<=^ is denoted by [2]. Received by the editors October 7, 1971. AMS 1970 subject classifications. Primary 12H05, 12E05; Secondary 13B25, 16A02.