Integral representations of cyclic groups of prime order
Integral representations of cyclic groups of prime order
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素数阶循环群的积分表示
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发表时间：
1957
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通讯作者：
I. Reiner
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作者：
I. Reiner
的话。在矩阵表示方面，这个引理暗示了R的不可约的^-表示类（在单模等价下）与P的理想类之间存在一一对应关系。通过将R的正则表示限制为R中的一个完整的不可约的矩阵表示集，特别地，设/(x)£Z[x]是不可约的，集合P = Z[0]，其中6是f(x)的零。由于R的每一个不可约表示都用6 - >x来描述，其中X是f(X) =0的非减积分解，因此这种矩阵解的单模类数与Z（[6]）的类数相一致。（参见[5；8]。）现在假设o是一个Dedekind环（见[4]），它被假设为一个规则的z模。根据引理1，每一个不可约的正则o模都与0中的理想o同构。
Remark. In terms of matrix representations, this lemma implies that there is a one-to-one correspondence between classes (under unimodular equivalence) of irreducible ^-representations of R and ideal classes of P. A full set of inequivalent irreducible matrix representations is obtained by restricting the regular representation of R to a full set of inequivalent ideals in R. In particular, let/(x)£Z[x] be irreducible, and set P = Z[0] where 6 is a zero of f(x). Since every irreducible representation of R is described by 6—>X, where X is an integral nonderogatory solution of f(X) =0, the number of unimodular classes of such matrix solutions coincides with the class number of Z[6]. (See [5; 8].) Now let o be a Dedekind ring (see [4]) which is assumed to be a regular Z-module. By Lemma 1, every irreducible regular o-module is o-isomorphic to an ideal in o.