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

Abstract

Let $\mathcal{O}$ denote the ring of integers of a quadratic field $\mathbb{Q}(\sqrt{-7})$. 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 $a_1{X}_1^2+a_2{X}_2^2+a_3{X}_3^2$ where $a_i\in \mathcal{O}$ for $1\le i\le 3$ for representing all $2\times 2$ matrices over $\mathcal{O}$. Let K denote a quadratic field such that its ring of integers ${\mathcal{O}}_K$ is a principal ideal domain and 2 is a product of two distinct primes. It is a well-known fact that $\mathbb{Q}(\sqrt{-7})$ is the only imaginary quadratic field with the above properties. Let $D_K$ denote the discriminant of K. We have $D_K\equiv 1(\text{mod }8)$ if and only if 2 is a product of two distinct primes in ${\mathcal{O}}_K$. With ${\mathcal{O}}_K$ as above, in this paper we generalize our earlier result. We give a necessary and sufficient condition for a diagonal quadratic form $\sum _{i=1}^m{a}_i{X}_i^2$ where $a_i\in {\mathcal{O}}_K$, $1\le i\le m$ to represent all $2\times 2$ matrices over ${\mathcal{O}}_K$. 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].

Keywords:

• ring of integers
• principal ideal domain
• diagonal quadratic form
• universality of a diagonal quadratic form
• prime element

AMS CLASSIFICATIONS:

• 11A05
• 11E25
• 15A24
• 15B33

Acknowledgements

We thank Professor S. A. Katre for informing us about the paper of Jungin Lee. The first author would like to thank Dr. Mukesh D. Patil and Dr. Arpita Palchowdhury for support and encouragement. The authors take the opportunity to thank Professor R. M. Pawale, Head, Department of Mathematics, University of Mumbai.

Disclosure statement

No potential conflict of interest was reported by the author(s).