The Ring of Integer-Valued Polynomials on 3 × 3 Matrices and Its Integral Closure

A polynomial f ( x ) ∈ ℚ [ x ] $$f(x) \in \mathbb {Q}[x]$$ is called integer-valued if f ( n ) ∈ ℤ $$f(n)\in \mathbb {Z}$$ for all n ∈ ℤ $$n\in \mathbb {Z}$$ . Bhargava’s p-orderings and p-sequences have been helpful tools in the study of integer-valued polynomials over subsets of ℤ $$\mathbb {Z}$$ and arbitrary Dedekind domains, and similar useful definitions exist of ν-orderings and ν-sequences in the case of certain noncommutative rings. In Evrard and Johnson (J Algebra 441:660–677, 2015), Evrard and Johnson use these ν-sequences to construct a regular p-local basis for the rational integer-valued polynomials over the integral closure of the ring of 2 × 2 integer matrices M 2 ( ℤ ) $$M_2(\mathbb {Z})$$ , by way of moving the problem to maximal orders within an index 2 division algebra over ℚ p $$\mathbb {Q}_p$$ . This paper demonstrates how the construction used there extends to maximal orders in index 3 division algebras over ℚ 2 $$\mathbb {Q}_2$$ , thereby giving the construction for a regular basis for polynomials that are integer-valued over this maximal order.