Tame Kernels of Quaternion Number Fields |
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Authors: | Haiyan Zhou Wenzhu Xie |
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Institution: | 1. School of Mathematical Sciences , Nanjing Normal University , Nanjing , China haiyanxiaodong@gmail.com;3. School of Mathematical Sciences , Nanjing Normal University , Nanjing , China |
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Abstract: | Let E/F be a Galois extension of number fields with the quaternion Galois group Q 8. In this paper, we prove some relations connecting orders of the odd part of the kernel of the transfer map of the tame kernel of E with the same orders of some of its subfields. Let E/? be a Galois extension of number fields with the Galois group Q 8 and p an odd prime such that p ≡ 3 (mod 4). We prove that if there is at most one quadratic subfield such that the p-Sylow subgroup of the tame kernel is nontrivial, then p r -rank(K 2(E/K)) is even, i.e., 2|p r -rank(K 2(𝒪 E )) ? p r -rank(K 2(𝒪 K )), where K is the quartic subfield of E. |
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Keywords: | Quaternion group Tame kernels Transfer map |
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