The 3-ranks of tame kernels of cubic cyclic number fields
The 3-ranks of tame kernels of cubic cyclic number fields
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DOI：
10.4064/aa129-4-7
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发表时间：
2007
期刊：
影响因子：
0.7
通讯作者：
Xuejun Guo
中科院分区：
文献类型：
--
作者：
Xuejun Guo
布朗金在1986年给出了两个证明。第一个证明是解析的，依赖于Mazur和Wiles的深度结果，而第二个证明是代数的，使用k理论中的精确序列。结合三次循环数域的三类群的相同精确序列和Gerth理论，我们可以处理具有任意多分枝素数的三次循环数域。本文的主要定理是定理4.4。由这个定理，可以得到一般三次循环数域的3秩公式。作为应用，我们在第4节中证明了以下定理。
Browkin gave two proofs in [1]. The first proof is analytic and depends on deep results by Mazur and Wiles, while the second one is algebraic, using an exact sequence in K-theory. In this paper, combining the same exact sequence and Gerth’s theory of the 3-class groups of cubic cyclic number fields, we can deal with cubic cyclic number fields with arbitrarily many ramified primes. The main theorem of this paper is Theorem 4.4. From this theorem, one can get the 3-rank formula for general cubic cyclic number fields. As an application, we prove the following theorem in Section 4.