On a family of quadratic fields whose class numbers are divisible by five
On a family of quadratic fields whose class numbers are divisible by five
复制标题
在类数可被 5 整除的二次域族上
DOI：
10.3792/pjaa.74.120
复制
发表时间：
1998
影响因子：
1.2
通讯作者：
Masahiko Sase
中科院分区：
文献类型：
--
作者：
Masahiko Sase
本文构造了一类类数可被5整除的二次域。我们推广了Kishi和Miyake[1]的方法，并利用了Kondo[2]引入的一个五次族，得到了这个结果。符号。在本文中，我们将使用以下符号。Z， Q将在通常意义上使用。对于一个有理数p和一个Z，一个v0，向上(a)将意味着最大的指数m，使得pm la。我们将考虑各种数域，即Q， k, k， L， F的有限扩展，如果p是素数理想，且a是整数理想4= 0在数域中，，(a)表示最大指数m，使得pm [a]。如果p是质数理想除p， e，/p表示p的分支指数。对于f(x) Z[x],f(J) (x)表示f(x)的第J阶导数。Cn表示n阶的环群；Dn是2n阶的二面体基团。hk表示数域k的类数，如果k是k的伽罗瓦扩展，则G （k /k）表示k /k的伽罗瓦群。1. 质数的分支。设q是一个奇素数，f(x)是q [x]中的一个不可约的q次多项式。设0是f(x)和f -Q(0)的根。我们用L表示f(x) / q的极小分裂域。我们首先证明命题1。设L ' Q] <2q，且f中没有素数完全分形，则G （L/Q）与D同构，L是L中包含的二次域k上Q次的非分形循环扩展，且唯一。证明。由于[L" Q] <2q, Q L [L" Q] G（L/Q）应为Cq或Dq。但是我们排除了C，因为我们假设了F/Q的分支。因此，G(L/Q) D，并且存在一个唯一的k，使得L k Q， [k" Q] 2和[L" k] Q。接下来，我们必须证明L/k是无歧化的。假设L的素数理想3的分支是L/k。由于L/k是q次的循环扩展，它的分支指数为q。由于[L ' F] 2，素数p3 （q F）为total ！y在F/Q中分叉。这与假设相矛盾。因为q是奇数，所以k的无穷素数也是非-
In this paper, we construct a family of quadratic fields whose class numbers are divisible by five. We obtain this result by extending the method of Kishi and Miyake [1] and using a family of quintics introduced by Kondo [2]. Notation. Throughout this paper, we shall use the following notation. Z, Q will be used in the usual sense. For a rational prime p and a Z, a ve 0, up (a) will mean the greatest exponent m such that pm la. We shall consider various number fields, i.e. finite extensions of Q, k, K, L, F, If p is a prime ideal and a an integral ideal 4= 0 in a number field, ,(a) will mean the greatest exponent m such that pm [a. If p is a prime ideal dividing p, e,/p will mean the ramification index of p. For f(x) Z[x] ,f(J) (x) will mean the jth derivative of f(x). Cn will mean the cyclic group with order n; Dn the dihedral group with order 2n. hk will mean the class number of a number field k. If K is a Galois extension of k, G (K/k)will mean the Galois group for K/k. 1. Ramification of primes. Let q be an odd prime and f(x) be an irreducible polynomial of degree q in Q[x]. Let 0 be a root of f(x) and F --Q(0). We denote by L the minimal splitting field of f(x) over Q. We shall first prove’ Proposition 1. Suppose L" Q] <2q and that no prime number is totally ramified in F. Then G (L/Q) is isomorphic to D and L is an unramifled cyclic extension of degree q over the quadratic field k contained in L which is unique. Proof. Since [L" Q] <2q and q l[L" Q] G(L/Q) should be Cq or Dq. But C is excluded because of our assumption on the ramification in F/Q. Thus G(L/Q) D and there is a unique k such that L k Q, [k" Q] 2 and [L" k] q. Next, we have to prove that L/k is unramified. Suppose a prime ideal 3 of L is ramified in L/k. Its ramification index is q since L/k is a cyclic extension with degree q. Since [L" F] 2, the prime p 3 (q F is tota!y ramified in F/Q. This contradicts to the assumption. Since q is odd, the infinite primes of k are also unrami-