Let p, q and <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1171_Article_IEq1.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$q'$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>q</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> be three distinct odd prime numbers, satisfying certain congruences. We give the structure of the unramified abelian Iwasawa module <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1171_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="106" /> </InlineMediaObject> <EquationSource Format="TEX">$X_{\infty }(\mathbb {Q}(\sqrt{pqq'}))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>X</mi> <mi>∞</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Q</mi> <mrow> <mo stretchy="false">(</mo> <msqrt> <mrow> <mi>p</mi> <mi>q</mi> <msup> <mi>q</mi> <mo>′</mo> </msup> </mrow> </msqrt> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of the number field <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1171_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {Q}(\sqrt{pqq'})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">(</mo> <msqrt> <mrow> <mi>p</mi> <mi>q</mi> <msup> <mi>q</mi> <mo>′</mo> </msup> </mrow> </msqrt> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. As an example, for <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1171_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="117" /> </InlineMediaObject> <EquationSource Format="TEX">$q=2^{82589933}-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>=</mo> <msup> <mn>2</mn> <mn>82589933</mn> </msup> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1171_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="127" /> </InlineMediaObject> <EquationSource Format="TEX">$q'=2^{136279841}-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>q</mi> <mo>′</mo> </msup> <mo>=</mo> <msup> <mn>2</mn> <mn>136279841</mn> </msup> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, the recently discovered prime numbers, we have: <Equation ID="Equ1"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1171_Article_Equ1.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="241" /> </MediaObject> <EquationSource Format="TEX">$ X_{\infty }(\mathbb {Q}(\sqrt{13qq'}))\simeq (\mathbb {Z}/2\mathbb {Z})^{2^{82589931}}. $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi>X</mi> <mi>∞</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Q</mi> <mrow> <mo stretchy="false">(</mo> <msqrt> <mrow> <mn>13</mn> <mi>q</mi> <msup> <mi>q</mi> <mo>′</mo> </msup> </mrow> </msqrt> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>≃</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">/</mo> <mn>2</mn> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mn>2</mn> <mn>82589931</mn> </msup> </msup> <mo>.</mo> </mrow> </math></EquationSource> </Equation>

On the unramified Abelian Iwasawa module of some number fields

期刊论文
发表日期：

2025年7月22日

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Abdelghani Assarrar,
Ali Mouhib

摘要

Let p, q and $q'$ $q^{'}$ be three distinct odd prime numbers, satisfying certain congruences. We give the structure of the unramified abelian Iwasawa module $X_{\infty }(\mathbb {Q}(\sqrt{pqq'}))$ $X_{\infty} (Q (\sqrt{p q q^{'}}))$ of the number field $\mathbb {Q}(\sqrt{pqq'})$ $Q (\sqrt{p q q^{'}})$ . As an example, for $q=2^{82589933}-1$ $q = 2^{82589933} - 1$ and $q'=2^{136279841}-1$ $q^{'} = 2^{136279841} - 1$ , the recently discovered prime numbers, we have: $ X_{\infty }(\mathbb {Q}(\sqrt{13qq'}))\simeq (\mathbb {Z}/2\mathbb {Z})^{2^{82589931}}. $ $X_{\infty} (Q (\sqrt{13 q q^{'}})) ≃ {(Z / 2 Z)}^{2^{82589931}} .$