<p>Let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ p\equiv -q\equiv -r\equiv 5 \pmod 8$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≡</mo> <mo>-</mo> <mi>q</mi> <mo>≡</mo> <mo>-</mo> <mi>r</mi> <mo>≡</mo> <mn>5</mn> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>8</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be pairwise different primes such that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\left( \dfrac{p}{q}\right) =\left( \dfrac{p}{r}\right) =1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfenced close=")" open="("> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> </mstyle> </mfenced> <mo>=</mo> <mfenced close=")" open="("> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>p</mi> <mi>r</mi> </mfrac> </mstyle> </mfenced> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we investigate the second 2-class group of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {K}_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">K</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>, the <i>n</i>th layer of the cyclotomic <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb {Z}_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-extension of the field <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathbb {K}:=\mathbb {Q}(\sqrt{pq}, \sqrt{pr})$</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">pq</mi> </mrow> </msqrt> <mo>,</mo> <msqrt> <mrow> <mi mathvariant="italic">pr</mi> </mrow> </msqrt> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. The capitulation problem is investigated too.</p>

Capitulation for the layers of certain cyclotomic $\mathbb {Z}_2$-extensions

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发表日期：

2026年5月11日

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Idriss Jerrari,
Abdellah Sbai,
Abdelmalek Azizi

摘要

Let $ p\equiv -q\equiv -r\equiv 5 \pmod 8$ $p \equiv - q \equiv - r \equiv 5 (mod 8)$ be pairwise different primes such that $\left( \dfrac{p}{q}\right) =\left( \dfrac{p}{r}\right) =1$ $(\frac{p}{q}) = (\frac{p}{r}) = 1$ . In this paper, we investigate the second 2-class group of $\mathbb {K}_n$ $K_{n}$ , the nth layer of the cyclotomic $\mathbb {Z}_2$ $Z_{2}$ -extension of the field $\mathbb {K}:=\mathbb {Q}(\sqrt{pq}, \sqrt{pr})$ $K : = Q (\sqrt{pq}, \sqrt{pr})$ . The capitulation problem is investigated too.

Capitulation for the layers of certain cyclotomic \(\mathbb {Z}_2\)-extensions

摘要