On Galois groups of local fields
On Galois groups of local fields
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关于局部域的伽罗瓦群
DOI：
10.1090/s0002-9947-1955-0075239-5
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发表时间：
1955
影响因子：
1.3
通讯作者：
K. Iwasawa
中科院分区：
文献类型：
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作者：
K. Iwasawa
在Q中作为基域，研究扩张Q/k的Galois群G(Q/k)的结构.设V是Q/k的分支域，即k在Q中的所有有限TAM分枝扩张的合成，G(2/V)和G(V/k)分别表示扩张Q/V和V/k的Galois群.我们将首先确定群G(V/k)=G(q/k)/G(q/V)和G(q2/V)的结构，并证明群扩张G(q/k)/G(q/V)是分裂的。因此，我们的主要结果是明确地描述了G(q/k)的内自同构对G(q/V)的模换位子群的因子群的影响，即对V在U中的极大交换扩张V‘的Galois群G(V’/V)的影响。这当然不足以完全确定群G(q/k)的结构；为此，我们还必须找到G(q/k)的内自同构对正规子群G(Q2/V)本身的影响。然而，它给了我们一些对G(q/k)的结构的洞察，我们希望它将在未来对G(q/k)群的研究以及对代数数域的Galois群的研究有所帮助。本文的提纲如下：在？1中，我们证明了一些群论引理，这些引理将在后面用到。在文献[2]中，我们研究了k的某类有限驯服分枝Galois扩张E在乘群E上作用的Galois群的性质，并利用这些结果证明了文献[3]中所提到的G(q/k)的性质。
in Q as the ground field and study the structure of the Galois group G(Q/k) of the extension Q/k. Let V be the ramification field of Q/k, i.e. the composite of all finite tamely ramified extensions of k in Q, and let G(2/ V) and G( V/k) denote the Galois groups of the extensions Q/ V and V/k respectively. We shall first determine the structure of the groups G(V/k) = G(Q/k)/G(Q/ V), and G(Q2/ V) and show that the group extension G(Q/k)/G(Q?/ V) splits. Our main result is, then, to describe explicitly the effect of inner automorphisms of G(Q/k) on the factor group of G(Q/ V) modulo its commutator subgroup, i.e., on the Galois group G( V'/ V) of the maximal abelian extension V' of V in U. This is, of course, not sufficient to determine the structure of the group G(Q/k) completely; to do that, we still have to find the effect of inner automorphisms of G(Q?/k) on the normal subgroup G(Q2/ V) itself. However, it gives us some insight into the structure of G(Q/k); and we hope it will help somehow, in the future, in the study of the group G(Q/k) as well as in that of the Galois groups of algebraic number fields. An outline of the paper is as follows: in ?1 we prove some group-theoretical lemmas which will be used later. In ?2 we study the behavior of the Galois group of a certain type of finite tamely ramified Galois extension E of k acting on the multiplicative group of E. Using those results, we then prove in ?3 the properties of G(Q/k) as mentioned above(').