<p>In this paper, we characterize all fields of the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {K}:=\mathbb {Q}(\sqrt{pr}, \sqrt{pq})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">K</mi> <mo>:</mo> <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">pq</mi> </mrow> </msqrt> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {L}:=\mathbb {Q}(\sqrt{pr}, \sqrt{pq}, \sqrt{2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">L</mi> <mo>:</mo> <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">pq</mi> </mrow> </msqrt> <mo>,</mo> <msqrt> <mn>2</mn> </msqrt> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> whose 2-class groups are cyclic, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ p\equiv 5 \pmod 8,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≡</mo> <mn>5</mn> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>8</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ q\equiv 3 \pmod 4 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>≡</mo> <mn>3</mn> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$r\equiv 3 \pmod 8$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>≡</mo> <mn>3</mn> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>8</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> are pairwise different primes. Furthermore, as application we study the cyclicity of the unramified Iwasawa module of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb {K}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">K</mi> </math></EquationSource> </InlineEquation>.</p>

Cyclicity of the unramified Iwasawa module of certain biquadratic number fields

期刊论文
发表日期：

2026年5月16日

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Abdelmalek Azizi,
Idriss Jerrari,
Abdellah Sbai

摘要

In this paper, we characterize all fields of the form $\mathbb {K}:=\mathbb {Q}(\sqrt{pr}, \sqrt{pq})$ $K : = Q (\sqrt{pr}, \sqrt{pq})$ and $\mathbb {L}:=\mathbb {Q}(\sqrt{pr}, \sqrt{pq}, \sqrt{2})$ $L : = Q (\sqrt{pr}, \sqrt{pq}, \sqrt{2})$ whose 2-class groups are cyclic, where $ p\equiv 5 \pmod 8,$ $p \equiv 5 (mod 8),$ $ q\equiv 3 \pmod 4 $ $q \equiv 3 (mod 4)$ and $r\equiv 3 \pmod 8$ $r \equiv 3 (mod 8)$ are pairwise different primes. Furthermore, as application we study the cyclicity of the unramified Iwasawa module of $\mathbb {K}$ $K$ .