Determination of all non-normal quartic CM-fields and of all non-abelian normal octic CM-fields with class number one
Determination of all non-normal quartic CM-fields and of all non-abelian normal octic CM-fields with class number one
复制标题
确定所有非正态四次 CM 场和所有第一类非阿贝尔正态八次 CM 场
DOI：
10.4064/aa-67-1-47-62
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发表时间：
1994
期刊：
影响因子：
0.7
通讯作者：
R. Okazaki
中科院分区：
文献类型：
--
作者：
S. Louboutin;R. Okazaki
0.导言和注释。人们早就知道，类数为1的非同构阿贝尔CM-域只有1000个。最近，K。Yamamura（见[Y]）确定了所有类数为1的阿贝尔CM-域。有172个非同构的这样的数域。这也是众所周知的。2])Hoffstein证明了类数为1的非同构正规CM-域的次数小于436（见[H，推论2]）。因此，现在是时候继续确定具有类数1和固定次数的非阿贝尔甚至非正规CM-域了。我们看的是最小的可能程度。指出不存在6次正规的非阿贝尔CM-场。因此，我们将研究非正规四次情形和八次非阿贝尔正规情形，即四元数或二面角情形。事实上，设N是在Q上正规的CM场。正如在[W，p.38]中，我们可以很容易地看到，复共轭是扩张N/Q的伽罗瓦群中的一个二阶元素，它与这个伽罗瓦群的所有其他元素交换。这个对正规CM-域的伽罗瓦群的约束使我们能够指出，例如，不存在任何非阿贝尔但正规的CM-域，其度为2 p，其中p是奇素数。因此，非阿贝尔正规CM场的最低可能度是8。这里我们知道，任何非交换正规八次数域的伽罗瓦群要么是四元数群，要么是二面体群。本文证明了不存在类数为1的四元数八次CM-域，因为这类数域的类数总是偶数。让我们注意到，在[娄4]的第一作者确定了所有四元数八次CM-域类数为2。然后我们证明，
0. Introduction and notations. It has long been known that there are only finitely many non-isomorphic abelian CM-fields with class number one. Lately, K. Yamamura (see [Y]) has determined all abelian CM-fields with class number one. There are exactly 172 non-isomorphic such number fields. It is also known (see [O, Th. 2]) that there are only finitely many non-isomorphic normal CM-fields with class number one and J. Hoffstein proved that the degrees of such fields are less than 436 (see [H, Corollary 2]). Hence, it is time to move on to the determination of non-abelian or even non-normal CM-fields with class number one and of fixed degrees. We look at the smallest possible degrees. We point out that there does not exist any non-abelian but normal CM-field with degree 6. Hence, we will look at the non-normal quartic case and at the octic non-abelian normal case, i.e. at the quaternion or dihedral cases. Indeed, let N be a CM-field that is normal over Q. As in [W, p. 38], one can easily see that the complex conjugation is an element of order two in the Galois group of the extension N/Q that commutes with all the other elements of this Galois group. This constraint on the Galois group of a normal CM-field enables us to point out that for example there does not exist any non-abelian but normal CM-field with degree 2p where p is an odd prime. Thus, the lowest possible degree for a non-abelian normal CM-field is 8. Here we know that the Galois group of any non-abelian normal octic number field is either a quaternion group or a dihedral group. In this paper, we prove that there does not exist any quaternion octic CM-field with class number one because the class number of such a number field is always even. Let us note that in [Lou 4] the first author determined all quaternion octic CM-fields with class number two. We then prove that there