Integer-valued Polynomials Over Matrix Rings of Number Fields

Abstract In this paper, we study the ring of integer-valued polynomials Int(Mn(OK )) := { f ∈ Mn(K)[x] | f (Mn(OK )) ⊆ Mn(OK )}, where K is a number field and OK is the ring of algebraic integers of K. We show that for a prime number p ∈ Z, the polynomial f p,n(x) := (x pn−x)(x pn−1−x)...(x p−x) p is an element of Int(Mn(OK )) if and only if p is a totally split prime in OK . Also, we consider the ring IntMn (Q)(Mn(OK )) := Int(Mn(OK )) Mn(Q)[x]. Then, we characterize finite Galois extensions K of Q in terms of the ring IntMn (Q)(Mn(OK )). In fact, we prove that IntMn (Q)(Mn(OK )) = IntMn (Q)(Mn(OK )) 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 IntMn (Q)(Mn(OK )). Then, we obtain many non-Noetherian integral domains, IntQ(OK ), between the ring Z[x] and the classical ring of integer-valued polynomials Int(Z).