Quasi-finite field

In mathematics, a quasi-finite field^[1] is a generalisation of a finite field. Standard local class field theory usually deals with complete valued fields whose residue field is finite (i.e., non-archimedean local fields), but the theory applies equally well when the residue field is only assumed quasi-finite.^[2]

Formal definition

A quasi-finite field is a perfect fieldK together with an isomorphism of topological groups

${\displaystyle \phi :{\hat {\mathbb {Z} }}\to \operatorname {Gal} (K_{s}/K),}$

where K_s is an algebraic closure of K (necessarily separable because K is perfect). The field extensionK_s/K is infinite, and the Galois group is accordingly given the Krull topology. The group ${\displaystyle {\widehat {\mathbb {Z} }}}$ is the profinite completion of integers with respect to its subgroups of finite index.

This definition is equivalent to saying that K has a unique (necessarily cyclic) extension K_n of degree n for each integer n ≥ 1, and that the union of these extensions is equal to K_s.^[3] Moreover, as part of the structure of the quasi-finite field, there is a generator F_n for each Gal(K_n/K), and the generators must be coherent, in the sense that if n divides m, the restriction of F_m to K_n is equal to F_n.

Examples

The most basic example, which motivates the definition, is the finite field K = GF(q). It has a unique cyclic extension of degree n, namely K_n = GF(qⁿ). The union of the K_n is the algebraic closure K_s. We take F_n to be the Frobenius element; that is, F_n(x) = x^q.

もう一つの例は、複素数体C上のTの形式ローラン級数の環であるK = C (( T ))である。(これらは単に形式的な冪級数であり、負の次数の項を有限個許容する。) このとき、Kは一意の巡回拡大を持つ。

${\displaystyle K_{n}=\mathbf {C} ((T^{1/n}))}$

の各n≥1についてn次の体で、その和集合はピュイズー級数体と呼ばれるKの代数的閉包であり、Gal( Kn _/ K )の生成元は

${\displaystyle F_{n}(T^{1/n})=e^{2\pi i/n}T^{1/n}。}$

この構成は、Cを任意の特性0の代数閉体Cに置き換えても成り立つ。^{[ 4 ]}

参照

• 擬有限体

注記

1. ( Artin & Tate 2009、§XI.3) は、この体が「守屋の公理」を満たすと述べている。
2. 守谷美香夫による記載 ( Serre 1979、第 XIII 章、p. 188)
3. ( Serre 1979、§XIII.2 演習 1、p. 192)
4. ( Serre 1979、§XIII.2、p. 191)

参考文献

• アルティン、エミール；テイト、ジョン（2009）[1967]、類体論、アメリカ数学会、ISBN 978-0-8218-4426-7、MR  2467155、Zbl  1179.11040
• セール、ジャン＝ピエール（1979）、局所場、大学院数学テキスト、第67巻、グリーンバーグ、マーヴィン・ジェイ訳、シュプリンガー・フェアラーク、ISBN 0-387-90424-7、MR  0554237、Zbl  0423.12016