Ron Evans , Franz Lemmermeyer , Zhi-Hong Sun , Mark Van Veen
{"title":"环类字段和哈塞结果","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":"{\"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}","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
摘要
对于无平方 d>1,让 M 表示 F=Q(-3d) 中阶 Z[-3d] 的环类域。哈塞证明,当且仅当存在一个 Q 的立方扩展 E,使得 E 和 F 具有相同的判别式时,3 平分 F 的类数。定义实立方根 v=(a+bd)1/3 和 v′=(a-bd)1/3,其中 a+bd 是 Q(d) 的基本单位。我们证明,当且仅当 v∈M 时,E 可以看作 Q(v+v′)。作为证明的副产品,我们给出了 a 和 b 的明确同余式,当且仅当 v∈M 时,这两个同余式成立,我们还证明了 F(v)/F 的相对判别式的规范位于{1,36}或{38,318},视 v∈M 或 v∉M 而定。然后,我们证明 v 总是在 F 的阶 Z[-27d] 的环类域中。上面的一些结果可以扩展到适当包含基本单元 a+bd 的 Q(d) 子集。
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 .