Extraction of mth roots in matrix rings over fields
Let F be a field with characteristic p (not necessarily positive), and \(M_ n(R)\) be the ring of \(n\times n\) matrices over R. The author concerns in this paper with the problem of establishing criteria to determine when the matrix equation \(X^ m=A\), \(A\in M_ n(F)\), has a solution in \(M_ n(F)\). It suffices to consider the case of a p(x)- primary matrix A, where p(x) is an irreducible polynomial over F. For this case, let \(e_ 1\geq e_ 2\geq..\). be the sequence of exponents of p(x) which form the elementary divisors of A, extended by adding infinitely many 0 terms. The criteria given in this paper fall into three cases: (1) \(p(x)=x:\) the criterion states that for all i, \(e_{(i-1)m+1}-e_{im}=0\) or 1. (2) p(x)\(\neq x\), m not divisible by p: if \(\beta\) is a root of the separable core q(x) of p(x) and \(E=F(\beta)\), 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 \(x^ m-\beta\). (3) p(x)\(\neq x\), m a power of p: the criterion combines the criteria in the first two cases: for all \(i\geq 1,e_{(i-1)m+1}-e_{im}=0\) or 1 and the multiplicity of each \(e_ i\) must be representable as a sum of integers, not necessarily distinct, each of which is a degree of some irreducible factor over E of the polynomial \(x^ m-\beta.\) Only some special cases of the above problem were considered earlier: \(F={\mathbb{C}}\) by \textit{J. H. M. Wedderburn} [Lectures on matrices (1964; Zbl 0121.261), pp. 119-122 and by \textit{F. R. Gantmacher} [Matrix theory, Vol. 1 (English translation 1959), pp. 232-235; see also the review of the Russian original (1953; Zbl 0050.248)], \(F={\mathbb{C}}\) and \(m=2\) by \textit{G. W. Cross} and \textit{P. Lancaster} [Linear Multilinear Algebra 1, 289- 293 (1974; Zbl 0283.15008)], \(F=GF(q)\) and \(A=I\), the identity matrix by \textit{J. H. Hodges} [Am. Math. Monthly 65, 518-520 (1958; Zbl 0082.244)].
scientific article; zbMATH DE number 4165080
scientific article