Matrices as a diagonal quadratic form over rings of integers of certain quadratic number fields

Abstract

Let denote the ring of integers of a quadratic field . In 2022, Murtuza and Garge [Murtuza N, Garge A. Universality of certain diagonal quadratic forms for matrices over a ring of integers, Indian Journal of Pure and Applied Mathematics, Published online; December 2022.] gave a necessary and sufficient condition for a diagonal quadratic form where for for representing all matrices over . Let K denote a quadratic field such that its ring of integers is a principal ideal domain and 2 is a product of two distinct primes. It is a well-known fact that is the only imaginary quadratic field with the above properties. Let denote the discriminant of K. We have if and only if 2 is a product of two distinct primes in . With as above, in this paper we generalize our earlier result. We give a necessary and sufficient condition for a diagonal quadratic form where , to represent all matrices over . This result is a conjecture stated in [Murtuza N, Garge A. Universality of certain diagonal quadratic forms for matrices over a ring of integers, Indian Journal of Pure and Applied Mathematics, Published online; December 2022].