A normal integral basis theorem
A normal integral basis theorem
复制标题
正规积分基定理
DOI：
10.1016/0021-8693(76)90065-x
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发表时间：
1976
影响因子：
0.9
通讯作者：
A. Fröhlich
中科院分区：
文献类型：
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作者：
A. Fröhlich
设N为有理数域Q上伽罗瓦群Gal (N/Q)= r并在其代数整数环上的正规代数数域（总是有限次）。形式为{A ~}的整数环oN在Z上的一组基，在Do中有一个固定的点，y穿过r，按照经典术语，它将被称为N的正规积分busis，一个众所周知的必要条件是N/Q被完全分叉（参见[12]）。然而，这是不够的。第一个与此相反的例子是由Martinet （cf.[1]）为8阶的四元数群发现的，随后（cf.[1, 21）证明了对于8阶和41阶的四元数群，E=-1 （mod 4）有无穷多个这样的例子。这里我们证明
Let N be a normal algebraic number field (always of finite degree) with Galois group Gal (N/Q)= r over the field Q of rational numbers and oN its ring of algebraic integers. A basis of oN over Z, the ring of integers, of form {a~}, with a fixed in Do and y running through r, will in accordance with classical terminology be called a normal integral busis of N. A well-known necessary condition for this to happen is that N/Q be tamely ramified (cf.[12]). This, however, is not sufficient. The first examples to the contrary were found by Martinet (cf.[ll]) for the quaternion group of order 8, and (cf.[l, 21) subsequently it was shown that there are infinitely many such examples for the quaternion groups of order 8 and of order 41*, E=-1 (mod 4). Here we prove