Let K denote an algebraic number field such that <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$n=[K:\mathbb {Q}]\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mi>K</mi> <mo>:</mo> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">]</mo> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, its ring of integers <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {O}_K$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">O</mi> <mi>K</mi> </msub> </math></EquationSource> </InlineEquation> is a principal ideal domain and 2 is a product of n distinct (non-associates) primes in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {O}_K$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">O</mi> <mi>K</mi> </msub> </math></EquationSource> </InlineEquation>. For quadratic field K with the above conditions, M. Nullwala and A. Garge [<CitationRef CitationID="CR3">3</CitationRef>] gave a necessary and sufficient condition for a diagonal quadratic form <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\displaystyle {\sum _{i=1}^{m}a_iX_i^2}$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <munderover> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <msub> <mi>a</mi> <mi>i</mi> </msub> <msubsup> <mi>X</mi> <mi>i</mi> <mn>2</mn> </msubsup> </mrow> </mstyle> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$a_i\in \mathcal {O}_K$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mi>i</mi> </msub> <mo>∈</mo> <msub> <mi mathvariant="script">O</mi> <mi>K</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$1\le i \le m$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>m</mi> </mrow> </math></EquationSource> </InlineEquation> for representing all <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$2\times 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> matrices over <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {O}_K$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">O</mi> <mi>K</mi> </msub> </math></EquationSource> </InlineEquation>. In this paper, we prove that the same necessary and sufficient condition for a diagonal quadratic form to represent all <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$2\times 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> matrices over <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathcal {O}_K$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">O</mi> <mi>K</mi> </msub> </math></EquationSource> </InlineEquation> also holds for an algebraic number field K such that <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$[K:\mathbb {Q}]>2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mi>K</mi> <mo>:</mo> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">]</mo> <mo>></mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.

Matrices over certain rings of integers with degree of extension greater than 2 as a diagonal quadratic form

期刊论文
发表日期：

2026年5月16日

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Murtuza Nullwala

摘要

Let K denote an algebraic number field such that $n=[K:\mathbb {Q}]\ge 2$ $n = [K : Q] \geq 2$ , its ring of integers $\mathcal {O}_K$ $O_{K}$ is a principal ideal domain and 2 is a product of n distinct (non-associates) primes in $\mathcal {O}_K$ $O_{K}$ . For quadratic field K with the above conditions, M. Nullwala and A. Garge [3] gave a necessary and sufficient condition for a diagonal quadratic form $\displaystyle {\sum _{i=1}^{m}a_iX_i^2}$ $\sum_{i = 1}^{m} a_{i} X_{i}^{2}$ where $a_i\in \mathcal {O}_K$ $a_{i} \in O_{K}$ for all $1\le i \le m$ $1 \leq i \leq m$ for representing all $2\times 2$ $2 \times 2$ matrices over $\mathcal {O}_K$ $O_{K}$ . In this paper, we prove that the same necessary and sufficient condition for a diagonal quadratic form to represent all $2\times 2$ $2 \times 2$ matrices over $\mathcal {O}_K$ $O_{K}$ also holds for an algebraic number field K such that $[K:\mathbb {Q}]>2$ $[K : Q] > 2$ .