Integer-valued Polynomials Over Matrix Rings of Number Fields

In this paper, we study the ring of integer-valued polynomials Int(Mn(O-K)) := {f is an element of M-n(K)[x] | f (M-n(O-K)) subset of M-n(O-K)}, where K is a number field and O-K is the ring of algebraic integers of K. We show that for a prime number p is an element of Z, the polynomial f(p,n)(x) := (xp(n) - x)(xp(n-1) - x) ... (x(p) - x) p is an element of Int(M-n(O-K)) if and only if p is a totally split prime in O-K. Also, we consider the ring Int(Mn (Q))(M-n(O-K)) := Int(M-n(O-K)) boolean AND M-n(Q)[x]. Then, we characterize finite Galois extensions K of Q in terms of the ring Int(Mn) ((Q))(M-n(O-K)). In fact, we prove that Int(Mn (Q))(M-n(O-K)) = Int(Mn (Q))(M-n(O-K')) if and only if K = K', where K, K' are two finite Galois extensions of Q. Finally, we present some results on Noetherian property of the rings Int(Mn (Q))(M-n(O-K)). Then, we obtain many non-Noetherian integral domains, IntQ(O-K), between the ring Z[x] and the classical ring of integer-valued polynomials Int(Z).