Ron Evans , Franz Lemmermeyer , Zhi-Hong Sun , Mark Van Veen
{"title":"Ring class fields and a result of Hasse","authors":"Ron Evans , Franz Lemmermeyer , Zhi-Hong Sun , Mark Van Veen","doi":"10.1016/j.jnt.2024.07.001","DOIUrl":null,"url":null,"abstract":"<div><p>For squarefree <span><math><mi>d</mi><mo>></mo><mn>1</mn></math></span>, let <em>M</em> denote the ring class field for the order <span><math><mi>Z</mi><mo>[</mo><msqrt><mrow><mo>−</mo><mn>3</mn><mi>d</mi></mrow></msqrt><mo>]</mo></math></span> in <span><math><mi>F</mi><mo>=</mo><mi>Q</mi><mo>(</mo><msqrt><mrow><mo>−</mo><mn>3</mn><mi>d</mi></mrow></msqrt><mo>)</mo></math></span>. Hasse proved that 3 divides the class number of <em>F</em> if and only if there exists a cubic extension <em>E</em> of <span><math><mi>Q</mi></math></span> such that <em>E</em> and <em>F</em> have the same discriminant. Define the real cube roots <span><math><mi>v</mi><mo>=</mo><msup><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><msqrt><mrow><mi>d</mi></mrow></msqrt><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></math></span> and <span><math><msup><mrow><mi>v</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>=</mo><msup><mrow><mo>(</mo><mi>a</mi><mo>−</mo><mi>b</mi><msqrt><mrow><mi>d</mi></mrow></msqrt><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></math></span>, where <span><math><mi>a</mi><mo>+</mo><mi>b</mi><msqrt><mrow><mi>d</mi></mrow></msqrt></math></span> is the fundamental unit in <span><math><mi>Q</mi><mo>(</mo><msqrt><mrow><mi>d</mi></mrow></msqrt><mo>)</mo></math></span>. We prove that <em>E</em> can be taken as <span><math><mi>Q</mi><mo>(</mo><mi>v</mi><mo>+</mo><msup><mrow><mi>v</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></math></span> if and only if <span><math><mi>v</mi><mo>∈</mo><mi>M</mi></math></span>. As byproducts of the proof, we give explicit congruences for <em>a</em> and <em>b</em> which hold if and only if <span><math><mi>v</mi><mo>∈</mo><mi>M</mi></math></span>, and we also show that the norm of the relative discriminant of <span><math><mi>F</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>/</mo><mi>F</mi></math></span> lies in <span><math><mo>{</mo><mn>1</mn><mo>,</mo><msup><mrow><mn>3</mn></mrow><mrow><mn>6</mn></mrow></msup><mo>}</mo></math></span> or <span><math><mo>{</mo><msup><mrow><mn>3</mn></mrow><mrow><mn>8</mn></mrow></msup><mo>,</mo><msup><mrow><mn>3</mn></mrow><mrow><mn>18</mn></mrow></msup><mo>}</mo></math></span> according as <span><math><mi>v</mi><mo>∈</mo><mi>M</mi></math></span> or <span><math><mi>v</mi><mo>∉</mo><mi>M</mi></math></span>. We then prove that <em>v</em> is always in the ring class field for the order <span><math><mi>Z</mi><mo>[</mo><msqrt><mrow><mo>−</mo><mn>27</mn><mi>d</mi></mrow></msqrt><mo>]</mo></math></span> in <em>F</em>. Some of the results above are extended for subsets of <span><math><mi>Q</mi><mo>(</mo><msqrt><mrow><mi>d</mi></mrow></msqrt><mo>)</mo></math></span> properly containing the fundamental units <span><math><mi>a</mi><mo>+</mo><mi>b</mi><msqrt><mrow><mi>d</mi></mrow></msqrt></math></span>.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24001677/pdfft?md5=4a76de3ef7096a558707691b3467bc3b&pid=1-s2.0-S0022314X24001677-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022314X24001677","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
For squarefree , let M denote the ring class field for the order in . Hasse proved that 3 divides the class number of F if and only if there exists a cubic extension E of such that E and F have the same discriminant. Define the real cube roots and , where is the fundamental unit in . We prove that E can be taken as if and only if . As byproducts of the proof, we give explicit congruences for a and b which hold if and only if , and we also show that the norm of the relative discriminant of lies in or according as or . We then prove that v is always in the ring class field for the order in F. Some of the results above are extended for subsets of properly containing the fundamental units .