Multiplicative Invariant Fields of Dimension ≦ 6.
Multiplicative Invariant Fields of Dimension ≦ 6.
复制标题
维数 ≤ 6 的乘法不变域。
DOI：
10.1090/memo/1403
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发表时间：
2023
影响因子：
1.9
通讯作者：
A.Yamasaki
中科院分区：
文献类型：
--
作者：
A.Hoshi;M.Kang;A.Yamasaki
的有限子群在Brown，Büllow，Neubüser，Wondratscheck，and Zassenhaus（1978）中被分类为共轭群;特别地，在中存在非共轭有限群。每个有限群都自然地成立，从而得到一个忠实格。这样，就有了这样的格子。给定a-格，群通过乘法作用作用于有理函数域，即纯单名自同构。我们关心的是固定域的合理性问题。我们的调查工具是菲尔德弗的非分支布劳尔群。众所周知，如果非分歧的Brauer群（记为）是非平凡的，则固定域在上不是有理的（=纯超越的）。Saltman在1990年发现了乘法不变域的非分歧Brauer群的一个公式。然而，计算一个特定的乘法不变域需要额外的努力，即使格的秩等于。有一个直接分解，其中是的某个子群。第一被加数，这是有关的忠实的线性表示，已被许多作者研究。但是第二个和并没有得到太多的关注，除了当排名是。定理1.在有限群中，设为与相关联的忠实格，则存在与的精确格。在这种情况下，因此。在Brown，Büllow，Neubüser，Wondratscheck，and Zassenhaus（1978）和差距Group（2008）中，这些群同构于，其GAP ID分别为（4，12，4，12），（4，32，1，2），（4，32，3，2），（4，33，3，1），（4，33，6，1）。定理2.存在（resp.）有限子群sin（resp.）设为具有秩的格（分别为）与每一个群体相关联。在这些格子中，精确地（分别）其中，满足条件的。相应组差距ID（实际上是CARAT ID）可以明确地确定。受这些结果的启发，我们构造了秩为，，（是任意正整数，且是任意奇素数）的格，满足且因此$\mathbb {C}（M）^
The finite subgroups ofare classified up to conjugation in Brown, Büllow, Neubüser, Wondratscheck, and Zassenhaus (1978); in particular, there existnon-conjugate finite groups in. Each finite groupofacts naturally on; thus we get a faithful-latticewith. In this way, there are exactlysuch lattices. Given a-latticewith, the groupacts on the rational function fieldby multiplicative actions, ie purely monomial automorphisms over. We are concerned with the rationality problem of the fixed field. A tool of our investigation is the unramified Brauer group of the fieldover. It is known that, if the unramified Brauer group, denoted by, is non-trivial, then the fixed fieldis not rational (= purely transcendental) over. A formula of the unramified Brauer groupfor the multiplicative invariant field was found by Saltman in 1990. However, to calculatefor a specific multiplicatively invariant field requires additional efforts, even when the latticeis of rank equal to. There is a direct decompositionwhereis some subgroup of. The first summand, which is related to the faithful linear representations of, has been investigated by many authors. But the second summanddoesn’t receive much attention except when the rank is. Theorem 1. Among thefinite groups, letbe the associated faithful-lattice with, there exist preciselylatticeswith. In these situations,and thus. Thegroups are isomorphic to,,,,whose GAP IDs are (4, 12, 4, 12),(4, 32, 1, 2),(4, 32, 3, 2),(4, 33, 3, 1),(4, 33, 6, 1) respectively in Brown, Büllow, Neubüser, Wondratscheck, and Zassenhaus (1978) and in The GAP Group (2008). Theorem 2. There exist(resp.) finite subgroupsin(resp.). Letbe the lattice with rank(resp.) associated to each group. Among these lattices precisely(resp.) of them satisfy the condition. The GAP IDs (actually the CARAT IDs) of the corresponding groupsmay be determined explicitly. Motivated by these results, we construct-latticesof rank,,(is any positive integer andis any odd prime number) satisfying thatand; and therefore $\mathbb {C}(M)^