Matrix powers over finite fields
Let \(GF(q)\) denote the finite field of order \(q=p^ e\), \(p\) odd, \(M\) denote the ring of \(2\times 2\) matrices over \(GF(q)\), and \(n\) denote a divisor of \(q-1\) such that \(n\geq 2\) and 4 doesn't divide \(n\). The authors consider the problem of determining the number of solutions \(T\) in \(M\) of the equation \(T^ n=B\) for given \(B\) in \(M\). Their work generalizes results obtained for \(n=2\) by \textit{T. P. Donovan} [Congr. Numerantium 66, 113-122 (1988; Zbl 0692.15009)]. The case where \(B\) is a scalar matrix and \(n\) is arbitrary was solved (for \(p=2\) as well as \(p\) odd) in the reviewer's paper on scalar matrix equations [Duke Math. J. 25, 291-296 (1958; Zbl 0082.021)]. When \(n=2\) and \(B=I\), the solutions \(T\) are called involutory matrices. For a detailed exposition of results on such matrices over finite commutative rings see \textit{B. R. McDonald} [Can. J. Math. 24, 369-378 (1972; Zbl 0278.15009)]. Also, as a related problem, the authors consider the lifting of solutions of \(T^ 2=B\) over Galois rings. The proofs in this paper are straight- forward and based on properties of minimal polynomials. Several notational problems to be aware of are (1) in formula (a) and possibly also (b), of Lemma 3, the factor \(q^ 2-q\) should be \(q^ 2+q\), (2) in the proof of Lemma 5, the given fixed matrix is denoted by \(T\) instead of \(B\) as earlier and (3) in line 1 and 4 of the proof of Theorem 6, \(T\) is mistakenly used instead of \(B\).
scientific article; zbMATH DE number 147584