<p>In this paper, we calculate the unit groups and the 2-class numbers of the fields <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1274_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="153" /> </InlineMediaObject> <EquationSource Format="TEX">$ \mathbb {K}= \mathbb {Q}(\sqrt{2}, \sqrt{p_1}, \sqrt{p_2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">K</mi> <mo>=</mo> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">(</mo> <msqrt> <mn>2</mn> </msqrt> <mo>,</mo> <msqrt> <msub> <mi>p</mi> <mn>1</mn> </msub> </msqrt> <mo>,</mo> <msqrt> <msub> <mi>p</mi> <mn>2</mn> </msub> </msqrt> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1274_Article_IEq2.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="195" /> </InlineMediaObject> <EquationSource Format="TEX">$ \mathbb {L}= \mathbb {Q}( \sqrt{-1},\sqrt{2}, \sqrt{p_1}, \sqrt{p_2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">L</mi> <mo>=</mo> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">(</mo> <msqrt> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msqrt> <mo>,</mo> <msqrt> <mn>2</mn> </msqrt> <mo>,</mo> <msqrt> <msub> <mi>p</mi> <mn>1</mn> </msub> </msqrt> <mo>,</mo> <msqrt> <msub> <mi>p</mi> <mn>2</mn> </msub> </msqrt> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1274_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$p_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>p</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1274_Article_IEq4.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$p_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>p</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> are two prime numbers satisfying <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1274_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="150" /> </InlineMediaObject> <EquationSource Format="TEX">$p_1\equiv p_2\equiv 1 \pmod {4}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>≡</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>≡</mo> <mn>1</mn> <mspace width="4.44443pt" /> <mrow> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

On the unit group of some multiquadratic fields

期刊论文
发表日期：

2025年8月4日

查看全文

Mohamed Mahmoud Chems-Eddin,
Hamza El Mamry

摘要

In this paper, we calculate the unit groups and the 2-class numbers of the fields $ \mathbb {K}= \mathbb {Q}(\sqrt{2}, \sqrt{p_1}, \sqrt{p_2})$ $K = Q (\sqrt{2}, \sqrt{p_{1}}, \sqrt{p_{2}})$ and $ \mathbb {L}= \mathbb {Q}( \sqrt{-1},\sqrt{2}, \sqrt{p_1}, \sqrt{p_2})$ $L = Q (\sqrt{- 1}, \sqrt{2}, \sqrt{p_{1}}, \sqrt{p_{2}})$ , where $p_1$ $p_{1}$ and $p_2$ $p_{2}$ are two prime numbers satisfying $p_1\equiv p_2\equiv 1 \pmod {4}$ $p_{1} \equiv p_{2} \equiv 1 (mod 4)$ .