For an integer <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$m\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, we aim to investigate the realizability of types of metacyclic-nonmodular groups, whose abelianization is <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {Z}/2 \mathbb {Z}\times \mathbb {Z}/2^m \mathbb {Z}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">/</mo> <mn>2</mn> <mi mathvariant="double-struck">Z</mi> <mo>×</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">/</mo> <msup> <mn>2</mn> <mi>m</mi> </msup> <mi mathvariant="double-struck">Z</mi> </mrow> </math></EquationSource> </InlineEquation>, as the Galois group of the maximal unramified 2-extension (resp. pro-2-extension) over certain number fields of 2-power degree (resp. cyclotomic <InlineEquation ID="IEq5"> <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>-extensions). Furthermore, we present some new techniques for studying Greenberg’s conjecture for some number fields. In particular, the reader can find results concerning the real quadratic fields <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$F=\mathbb {Q}(\sqrt{\eta q rs})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mo>=</mo> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">(</mo> <msqrt> <mrow> <mi>η</mi> <mi>q</mi> <mi>r</mi> <mi>s</mi> </mrow> </msqrt> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, the real biquadratic fields <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$K=\mathbb {Q}(\sqrt{\eta q},\sqrt{rs})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>=</mo> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">(</mo> <msqrt> <mrow> <mi>η</mi> <mi>q</mi> </mrow> </msqrt> <mo>,</mo> <msqrt> <mrow> <mi mathvariant="italic">rs</mi> </mrow> </msqrt> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, with <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\eta \in \{1,2\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>η</mi> <mo>∈</mo> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>, and the Fröhlich multiquadratic fields of the form <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathbb {F}=\mathbb {Q}(\sqrt{q }, \sqrt{r}, \sqrt{s})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">F</mi> <mo>=</mo> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">(</mo> <msqrt> <mi>q</mi> </msqrt> <mo>,</mo> <msqrt> <mi>r</mi> </msqrt> <mo>,</mo> <msqrt> <mi>s</mi> </msqrt> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where q, r and s are odd prime numbers.

On the existence of the maximal unramified pro-2-extension over the cyclotomic $\mathbb {Z}_2$-extension with prescribed metacyclic Galois group

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发表日期：

2026年5月7日

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Mohamed Mahmoud Chems-Eddin,
Hamza El Mamry

摘要

For an integer $m\ge 2$ $m \geq 2$ , we aim to investigate the realizability of types of metacyclic-nonmodular groups, whose abelianization is $\mathbb {Z}/2 \mathbb {Z}\times \mathbb {Z}/2^m \mathbb {Z}$ $Z / 2 Z \times Z / 2^{m} Z$ , as the Galois group of the maximal unramified 2-extension (resp. pro-2-extension) over certain number fields of 2-power degree (resp. cyclotomic $\mathbb {Z}_2$ $Z_{2}$ -extensions). Furthermore, we present some new techniques for studying Greenberg’s conjecture for some number fields. In particular, the reader can find results concerning the real quadratic fields $F=\mathbb {Q}(\sqrt{\eta q rs})$ $F = Q (\sqrt{η q r s})$ , the real biquadratic fields $K=\mathbb {Q}(\sqrt{\eta q},\sqrt{rs})$ $K = Q (\sqrt{η q}, \sqrt{rs})$ , with $\eta \in \{1,2\}$ $η \in {1, 2}$ , and the Fröhlich multiquadratic fields of the form $\mathbb {F}=\mathbb {Q}(\sqrt{q }, \sqrt{r}, \sqrt{s})$ $F = Q (\sqrt{q}, \sqrt{r}, \sqrt{s})$ , where q, r and s are odd prime numbers.

On the existence of the maximal unramified pro-2-extension over the cyclotomic \(\mathbb {Z}_2\)-extension with prescribed metacyclic Galois group

摘要