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Application of Greenberg's conjecture in the capitulation problem

机译：格林伯格猜想在投降问题中的应用

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Abstract > Let r be a positive integer. Assume Greenberg's conjecture for some totally real number fields, we show that there exists an infinite family of imaginary cyclic number fields F over the field of rational number field <mat:math display="inline" altimg="urn:x-wiley:0025584X:media:mana201700467:mana201700467-math-0002" wiley:location="equation/mana201700467-math-0002.png"> <mat:mi mathvariant="double-struck">Q</mat:mi> </mat:math> , with an elementary 2‐class group of rank equal to r that capitulates in an unramified quadratic extension over F . Also, we give necessary and sufficient conditions for the Galois group of the unramified maximal 2‐extension over F to be abelian. </abstract> </span> <span class="z_kbtn z_kbtnclass hoverxs" style="display: none;">展开▼</span> </div> <div class="translation abstracttxt"> <span class="zhankaihshouqi fivelineshidden" id="abstract"> <span>机译：</span><Abstract Type =“Main”XML：Lang =“en”> <标题类型=“main”>抽象</ title> > 让 R </ I> 是一个正整数。假设Greenberg的猜想为一些完全实数的字段，我们表明存在无限的虚构循环数字段 f </ i> 在理性数字字段领域 <MAT：数学显示=“内联”Altimg =“URN：X-Wiley：0025584X：MANA201700467：MANA201700467 -MANA201700467-MATH-0002”WIley：Location =“等式/ MANA201700467-MATH-0002.PNG”> <mat：mi mathvariant =“双击”> q </ mat：mi> </ mat：math> ，基本的2级等级等于 R </ I> 该提议在未经修改的二次延伸中 f </ i> 。此外，我们对未经修改的最大2延伸的Galois组给予必要和充分的条件 f </ i> 是阿贝利安。 </ p> </摘要> </span> <span class="z_kbtn z_kbtnclass hoverxs" style="display: none;">展开▼</span> </div> </div> <div class="record"> <h2 class="all_title" id="enpatent33" >著录项</h2> <ul> <li> <span class="lefttit">来源</span> <div style="width: 86%;vertical-align: text-top;display: inline-block;"> <a href='/journal-foreign-24615/'>《Mathematische Nachrichten》</a> <b style="margin: 0 2px;">|</b><span>2019年第3期</span><b style="margin: 0 2px;">|</b><span>共7页</span> </div> </li> <li> <div class="author"> <span class="lefttit">作者</span> <p id="fAuthorthree" class="threelineshidden zhankaihshouqi"> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Mouhib Ali&option=202" target="_blank" rel="nofollow">Mouhib Ali;</a> </p> <span class="z_kbtnclass z_kbtnclassall hoverxs" id="zkzz" style="display: none;">展开▼</span> </div> </li> <li> <div style="display: flex;"> <span class="lefttit">作者单位</span> <div style="position: relative;margin-left: 3px;max-width: 639px;"> <div class="threelineshidden zhankaihshouqi" id="fOrgthree"> <p>Department of MathematicsKing Khalid UniversityP.O. Box 9004 Abha Saudi Arabia;</p> </div> <span class="z_kbtnclass z_kbtnclassall hoverxs" id="zhdw" style="display: none;">展开▼</span> </div> </div> </li> <li > <span class="lefttit">收录信息</span> <span style="width: 86%;vertical-align: text-top;display: inline-block;"></span> </li> <li> <span class="lefttit">原文格式</span> <span>PDF</span> </li> <li> <span class="lefttit">正文语种</span> <span>eng</span> </li> <li> <span class="lefttit">中图分类</span> <span><a href="https://www.zhangqiaokeyan.com/clc/156.html" title="数学">数学;</a></span> </li> <li class="antistop"> <span class="lefttit">关键词</span> <p style="width: 86%;vertical-align: text-top;"> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=capitulation problem&option=203" rel="nofollow">capitulation problem;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=class group&option=203" rel="nofollow">class group;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Iwasawa theory of Z 2 ‐extensions&option=203" rel="nofollow">Iwasawa theory of Z 2 ‐extensions;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=unit group&option=203" rel="nofollow">unit group;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=11R23&option=203" rel="nofollow">11R23;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=11R29&option=203" rel="nofollow">11R29;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=11R32&option=203" rel="nofollow">11R32;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=11R37&option=203" rel="nofollow">11R37;</a> </p> <div class="translation"> 机译：投入问题;班级组;Z 2 -extensions的iwasawa理论;单位组;11 r23;11r29;11r32;11r37; 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