{"title":"Small generators of abelian number fields","authors":"Martin Widmer","doi":"10.1515/forum-2023-0467","DOIUrl":null,"url":null,"abstract":"We show that for each abelian number field <jats:italic>K</jats:italic> of sufficiently large degree <jats:italic>d</jats:italic> there exists an element <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>α</m:mi> <m:mo>∈</m:mo> <m:mi>K</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0467_eq_0135.png\" /> <jats:tex-math>{\\alpha\\in K}</jats:tex-math> </jats:alternatives> </jats:inline-formula> with <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>K</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mi>ℚ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>α</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0467_eq_0100.png\" /> <jats:tex-math>{K=\\mathbb{Q}(\\alpha)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and absolute Weil height <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mi>H</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>α</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:msub> <m:mo>≪</m:mo> <m:mi>d</m:mi> </m:msub> <m:msup> <m:mrow> <m:mo stretchy=\"false\">|</m:mo> <m:msub> <m:mi mathvariant=\"normal\">Δ</m:mi> <m:mi>K</m:mi> </m:msub> <m:mo stretchy=\"false\">|</m:mo> </m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mrow> <m:mn>2</m:mn> <m:mo></m:mo> <m:mi>d</m:mi> </m:mrow> </m:mfrac> </m:msup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0467_eq_0091.png\" /> <jats:tex-math>{H(\\alpha)\\ll_{d}|\\Delta_{K}|^{\\frac{1}{2d}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi mathvariant=\"normal\">Δ</m:mi> <m:mi>K</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0467_eq_0128.png\" /> <jats:tex-math>{\\Delta_{K}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> denotes the discriminant of <jats:italic>K</jats:italic>. This answers a question of Ruppert from 1998 in the case of abelian extensions of sufficiently large degree. We also show that the exponent <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mfrac> <m:mn>1</m:mn> <m:mrow> <m:mn>2</m:mn> <m:mo></m:mo> <m:mi>d</m:mi> </m:mrow> </m:mfrac> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0467_eq_0152.png\" /> <jats:tex-math>{\\frac{1}{2d}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is best-possible when <jats:italic>d</jats:italic> is even.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Forum Mathematicum","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/forum-2023-0467","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We show that for each abelian number field K of sufficiently large degree d there exists an element α∈K{\alpha\in K} with K=ℚ(α){K=\mathbb{Q}(\alpha)} and absolute Weil height H(α)≪d|ΔK|12d{H(\alpha)\ll_{d}|\Delta_{K}|^{\frac{1}{2d}}}, where ΔK{\Delta_{K}} denotes the discriminant of K. This answers a question of Ruppert from 1998 in the case of abelian extensions of sufficiently large degree. We also show that the exponent 12d{\frac{1}{2d}} is best-possible when d is even.
我们证明,对于每个阶数为 d 的无性数域 K,都存在一个元素 α∈K {alpha\in K} ,其中 K = ℚ ( α ) {K=\mathbb{Q}(\alpha)} 且绝对韦尔高 H ( α ) ≪ d | Δ K | 1 2 d {H(\alpha)\ll_{d}|\Delta_{K}|^{\frac{1}{2d}} ,其中 Δ K {Delta_{K}} 表示 K 的判别式。 其中 Δ K {Delta_{K}} 表示 K 的判别式。这回答了鲁珀特在 1998 年提出的一个问题,即在阶数足够大的无性扩展的情况下。我们还证明了当 d 为偶数时,指数 1 2 d {\frac{1}{2d}} 是最可能的。
期刊介绍:
Forum Mathematicum is a general mathematics journal, which is devoted to the publication of research articles in all fields of pure and applied mathematics, including mathematical physics. Forum Mathematicum belongs to the top 50 journals in pure and applied mathematics, as measured by citation impact.