Extraction of mth roots in matrix rings over fields

AbstractA matrix A ϵ Mn(F), F an arbitrary field with characteristic p (not necessarily positive), is an mth power in Mn(F) if and only if each of its p(x)-primary components is, where p(x) runs through the irreducible factors of the minimal or characteristic polynomial of A. This paper establishes necessary and sufficient criteria for determining when such a p(x)-primary matrix is an mth power. The criteria fall into three cases: (1) p(x) = x: If e1 ⩾ e2 ⩾ … is the sequence of exponents of p(x) which form the elementary divisors of A, extended by adding infinitely many 0 terms, the criterion states that for all i ⩾ 1, e(i−1)m+1−eim = 0 or 1. (2) p(x) ≠ x, m not divisible by p: If β is a root of the separable core q(x) of p(x) and E = F(β), then the multiplicity of each elementary divisor of A must be representable as a sum of integers, not necessarily distinct, each of which is the degree of some irreducible factor over E of the polynomial xm − β. (3) p(x) ≠ x, m a power of p: In this case, the criterion combines the criteria in the first two cases: If e1 ⩾ e2 ⩾ … is the sequence of exponents of p(x) which form the elementary divisors of A, extended by adding infinitely many 0 terms, then for all i ⩾ 1, e(i−1)m+1 − eim = 0 or 1 and the multiplicity of each ei must be representable as a sum of integers, not necessarily distinct, each of which is the degree of some irreducible factor over E of the polynomial xm−β