The group of unramified Kummer extensions of prime degree
The group of unramified Kummer extensions of prime degree
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素次无枝库默扩张群
DOI：
10.1112/plms/s3-35.3.407
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发表时间：
1977
影响因子：
1.8
通讯作者：
L. Childs
中科院分区：
文献类型：
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作者：
L. Childs
设R是交换环，G是有限交换群.本文研究了Harrison [9]群Gal（i？，G）与群G的可分Galois扩张的同构类（在[1，2]的意义下）。将伽罗瓦扩张看作群环R [G]上的模，从Gal（j？，G）到Pic（jR [6 r]），秩一射影i？[Cr]-模，其核NB（i2，G）是具有正规基的Galois扩张群.在[6]中，把映射i称为Picard不变映射。本文描述了当G是素数阶循环的I时，i的象和核，其中I不是R中的零因子，且R包含一个本原的Z次单位根ε.
Let R be a commutative ring, and let G be a finite abelian group. This paper is concerned with the Harrison [9] group Gal (i?, G) of isomorphism classes of separable Galois extensions of R with group G (in the sense of [1, 2]). Viewing a Galois extension as a module over the group ring R [G] yields a homomorphism i from Gal (j?, G) to Pic (jR [6r]), the group of isomorphism classes of rank-one projective i?[Cr]-modules, whose kernel NB (i2, G) is the group of Galois extensions with normal basis. The map i was called the Picard invariant map in [6]. In this paper we describe both the image and the kernel of i when G is cyclic of prime order I, where I is not a zero divisor in R and R contains a primitive Zth root of unity£.