Let K be a number field with ring of integers <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\mathcal {O}}_K.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">O</mi> <mi>K</mi> </msub> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\mathcal {N}}_K$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">N</mi> <mi>K</mi> </msub> </math></EquationSource> </InlineEquation> be the set of positive integers n such that there exist units <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varepsilon , \delta \in {\mathcal {O}}_K^\times $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>,</mo> <mi>δ</mi> <mo>∈</mo> <msubsup> <mi mathvariant="script">O</mi> <mi>K</mi> <mo>×</mo> </msubsup> </mrow> </math></EquationSource> </InlineEquation> satisfying <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\varepsilon + \delta = n.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>+</mo> <mi>δ</mi> <mo>=</mo> <mi>n</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> We show that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">${\mathcal {N}}_K$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">N</mi> <mi>K</mi> </msub> </math></EquationSource> </InlineEquation> is a finite set if K does not contain any real quadratic subfield. In the case where K is a cubic field, we also explicitly classify all solutions to the unit equation <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\varepsilon + \delta = n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>+</mo> <mi>δ</mi> <mo>=</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> when K is either cyclic or has negative discriminant.

Sums of two units in number fields

期刊论文
发表日期：

2026年1月9日

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Magdaléna Tinková,
Robin Visser,
Pavlo Yatsyna

摘要

Let K be a number field with ring of integers ${\mathcal {O}}_K.$ $O_{K} .$ Let ${\mathcal {N}}_K$ $N_{K}$ be the set of positive integers n such that there exist units $\varepsilon , \delta \in {\mathcal {O}}_K^\times $ $ε, δ \in O_{K}^{\times}$ satisfying $\varepsilon + \delta = n.$ $ε + δ = n .$ We show that ${\mathcal {N}}_K$ $N_{K}$ is a finite set if K does not contain any real quadratic subfield. In the case where K is a cubic field, we also explicitly classify all solutions to the unit equation $\varepsilon + \delta = n$ $ε + δ = n$ when K is either cyclic or has negative discriminant.