التفاصيل البيبلوغرافية
| العنوان: |
Ring class fields and a result of Hasse. |
| المؤلفون: |
Evans, Ron1 (AUTHOR) revans@ucsd.edu, Lemmermeyer, Franz2 (AUTHOR) hb3@uni-heidelberg.de, Sun, Zhi-Hong3 (AUTHOR) zhsun@hytc.edu.cn, Van Veen, Mark4 (AUTHOR) mavanveen@ucsd.edu |
| المصدر: |
Journal of Number Theory. Jan2025, Vol. 266, p33-61. 29p. |
| مصطلحات موضوعية: |
*CUBES, *SIGNS & symbols |
| مستخلص: |
For squarefree d > 1 , let M denote the ring class field for the order Z [ − 3 d ] in F = Q (− 3 d). 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 + b d) 1 / 3 and v ′ = (a − b d) 1 / 3 , where a + b d is the fundamental unit in Q (d). We prove that E can be taken as Q (v + v ′) if and only if v ∈ M. As byproducts of the proof, we give explicit congruences for a and b which hold if and only if v ∈ M , and we also show that the norm of the relative discriminant of F (v) / F lies in { 1 , 3 6 } or { 3 8 , 3 18 } according as v ∈ M or v ∉ M. We then prove that v is always in the ring class field for the order Z [ − 27 d ] in F. Some of the results above are extended for subsets of Q (d) properly containing the fundamental units a + b d. [ABSTRACT FROM AUTHOR] |
| قاعدة البيانات: |
Academic Search Index |