大次非交换数域的类数
Class numbers of large degree nonabelian number fields
摘要：
当数域的度和判别式都很大时，类数的计算变得相当困难，特别是在没有GRH假设的情况下。本文将无条件地证明某个120次非交换数域的类数为1。这个域是有理数唯一的$A_5 \times C_2$扩张，它只在653处分支，分歧指数为2。它是无条件证明类数为类数1的最大度数域。证明使用Guàrdia，Montes和Nart的算法计算一个整基，然后找到小素数幂范数的整元，建立类数的上界;进一步的代数论证证明了类数为1。这些技巧也可以应用于其他非阿贝尔数域。
英文摘要：
If a number field has a large degree and discriminant, the computation of the class number becomes quite difficult, especially without the assumption of GRH. In this article, we will unconditionally show that a certain nonabelian number field of degree 120 has class number one. This field is the unique $A_5 \times C_2$ extension of the rationals that is ramified only at 653 with ramification index 2. It is the largest degree number field unconditionally proven to have class number 1. The proof uses the algorithm of Guàrdia, Montes and Nart to calculate an integral basis and then finds integral elements of small prime power norm to establish an upper bound for the class number; further algebraic arguments prove the class number is 1. It is possible to apply these techniques to other nonabelian number fields as well.
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