<p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40316_2025_248_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="81" /> </InlineMediaObject> <EquationSource Format="TEX">$\Gamma ={\mathbb {Q}}(\root 3 \of {n})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Γ</mi> <mo>=</mo> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">(</mo> <mroot> <mi>n</mi> <mn>3</mn> </mroot> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be a pure cubic field with normal closure <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40316_2025_248_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="102" /> </InlineMediaObject> <EquationSource Format="TEX">$k={\mathbb {Q}}(\root 3 \of {n},\zeta ),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">(</mo> <mroot> <mi>n</mi> <mn>3</mn> </mroot> <mo>,</mo> <mi>ζ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40316_2025_248_Article_IEq3.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$n>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> denotes a cube free integer, and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40316_2025_248_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\zeta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ζ</mi> </math></EquationSource> </InlineEquation> is a primitive cube root of unity. Suppose <i>k</i> possesses an elementary bicyclic 3-class group <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40316_2025_248_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">${\textrm{Cl}}_3(k),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>Cl</mtext> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> and the conductor of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40316_2025_248_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$k/{\mathbb {Q}}(\zeta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo stretchy="false">/</mo> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">(</mo> <mi>ζ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> has the shape <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40316_2025_248_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="154" /> </InlineMediaObject> <EquationSource Format="TEX">$f\in \lbrace pq_1q_2,3pq,9pq\rbrace $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <mo stretchy="false">{</mo> <mi>p</mi> <msub> <mi>q</mi> <mn>1</mn> </msub> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>,</mo> <mn>3</mn> <mi>p</mi> <mi>q</mi> <mo>,</mo> <mn>9</mn> <mi>p</mi> <mi>q</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40316_2025_248_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="99" /> </InlineMediaObject> <EquationSource Format="TEX">$p\equiv 1\,({\textrm{mod}}\,9)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≡</mo> <mn>1</mn> <mspace width="0.166667em" /> <mo stretchy="false">(</mo> <mtext>mod</mtext> <mspace width="0.166667em" /> <mn>9</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40316_2025_248_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="157" /> </InlineMediaObject> <EquationSource Format="TEX">$q,q_1,q_2\equiv 2,5\,({\textrm{mod}}\,9)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>,</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>≡</mo> <mn>2</mn> <mo>,</mo> <mn>5</mn> <mspace width="0.166667em" /> <mrow> <mo stretchy="false">(</mo> <mtext>mod</mtext> <mspace width="0.166667em" /> <mn>9</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are primes. It is disproved that there are only two possible capitulation types <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40316_2025_248_Article_IEq10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$\varkappa (k),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϰ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> either type <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40316_2025_248_Article_IEq11.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">${\textrm{a}}.1,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>a</mtext> <mo>.</mo> <mn>1</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> (0000),  or type <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40316_2025_248_Article_IEq12.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">${\textrm{a}}.2,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>a</mtext> <mo>.</mo> <mn>2</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> (1000). Evidence is provided, theoretically and experimentally, of two further types, <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40316_2025_248_Article_IEq13.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">${\text {b}}.10,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>b</mtext> <mo>.</mo> <mn>10</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> (0320),  and <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40316_2025_248_Article_IEq14.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">${\textrm{d}}.23,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>d</mtext> <mo>.</mo> <mn>23</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> (1320).</p>

The capitulation problem in certain pure cubic fields

期刊论文
发表日期：

2025年9月12日

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Siham Aouissi,
Daniel C. Mayer

摘要

Let $\Gamma ={\mathbb {Q}}(\root 3 \of {n})$ $Γ = Q (\sqrt[3]{n})$ be a pure cubic field with normal closure $k={\mathbb {Q}}(\root 3 \of {n},\zeta ),$ $k = Q (\sqrt[3]{n}, ζ),$ where $n>1$ $n > 1$ denotes a cube free integer, and $\zeta $ $ζ$ is a primitive cube root of unity. Suppose k possesses an elementary bicyclic 3-class group ${\textrm{Cl}}_3(k),$ ${Cl}_{3} (k),$ and the conductor of $k/{\mathbb {Q}}(\zeta )$ $k / Q (ζ)$ has the shape $f\in \lbrace pq_1q_2,3pq,9pq\rbrace $ $f \in {p q_{1} q_{2}, 3 p q, 9 p q}$ where $p\equiv 1\,({\textrm{mod}}\,9)$ $p \equiv 1 (mod 9)$ and $q,q_1,q_2\equiv 2,5\,({\textrm{mod}}\,9)$ $q, q_{1}, q_{2} \equiv 2, 5 (mod 9)$ are primes. It is disproved that there are only two possible capitulation types $\varkappa (k),$ $ϰ (k),$ either type ${\textrm{a}}.1,$ $a . 1,$ (0000), or type ${\textrm{a}}.2,$ $a . 2,$ (1000). Evidence is provided, theoretically and experimentally, of two further types, ${\text {b}}.10,$ $b . 10,$ (0320), and ${\textrm{d}}.23,$ $d . 23,$ (1320).