环类字段和哈塞结果

Pub Date : 2024-08-20 DOI:10.1016/j.jnt.2024.07.001
Ron Evans , Franz Lemmermeyer , Zhi-Hong Sun , Mark Van Veen
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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>. 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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>. 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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。
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Ring class fields and a result of Hasse

For squarefree d>1, let M denote the ring class field for the order Z[3d] in F=Q(3d). Hasse proved that 3 divides the class number of F if and only if there exists a cubic extension E of Q such that E and F have the same discriminant. Define the real cube roots v=(a+bd)1/3 and v=(abd)1/3, where a+bd is the fundamental unit in Q(d). We prove that E can be taken as Q(v+v) if and only if vM. As byproducts of the proof, we give explicit congruences for a and b which hold if and only if vM, and we also show that the norm of the relative discriminant of F(v)/F lies in {1,36} or {38,318} according as vM or vM. We then prove that v is always in the ring class field for the order Z[27d] in F. Some of the results above are extended for subsets of Q(d) properly containing the fundamental units a+bd.

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