In this paper, we investigate the unit group of the real triquadratic number field <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1224_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="162" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {K}}_{1}={\mathbb {Q}}(\sqrt{2},\sqrt{pr}, \sqrt{qr} )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">K</mi> <mn>1</mn> </msub> <mo>=</mo> <mi mathvariant="double-struck">Q</mi> <mrow> <mo stretchy="false">(</mo> <msqrt> <mn>2</mn> </msqrt> <mo>,</mo> <msqrt> <mrow> <mi mathvariant="italic">pr</mi> </mrow> </msqrt> <mo>,</mo> <msqrt> <mrow> <mi mathvariant="italic">qr</mi> </mrow> </msqrt> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where p, q and r are three distinct primes satisfying certain conditions. Furthermore, we study the Iwasawa invariants of the cyclotomic <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1224_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {Z}}_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-extension of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1224_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="126" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {K}}={\mathbb {Q}}(\sqrt{pr}, \sqrt{qr})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">K</mi> <mo>=</mo> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">(</mo> <msqrt> <mrow> <mi mathvariant="italic">pr</mi> </mrow> </msqrt> <mo>,</mo> <msqrt> <mrow> <mi mathvariant="italic">qr</mi> </mrow> </msqrt> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.

Unit group of some triquadratic number fields and Greenberg’s conjecture

期刊论文
发表日期：

2025年5月6日

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Abdellah Sbai,
Idriss Jerrari,
Abdelmalek Azizi

摘要

In this paper, we investigate the unit group of the real triquadratic number field ${\mathbb {K}}_{1}={\mathbb {Q}}(\sqrt{2},\sqrt{pr}, \sqrt{qr} )$ $K_{1} = Q (\sqrt{2}, \sqrt{pr}, \sqrt{qr})$ , where p, q and r are three distinct primes satisfying certain conditions. Furthermore, we study the Iwasawa invariants of the cyclotomic ${\mathbb {Z}}_2$ $Z_{2}$ -extension of ${\mathbb {K}}={\mathbb {Q}}(\sqrt{pr}, \sqrt{qr})$ $K = Q (\sqrt{pr}, \sqrt{qr})$ .