In mathematics, a quasi-finite field 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.
Formal definition A quasi-finite field is a perfect field K together with an isomorphism of topological groups
ϕ : Z ^ → Gal ( K s / K ) , {\displaystyle \phi :{\hat {\mathbb {Z} }}\to \operatorname {Gal} (K_{s}/K),}
where Ks is an algebraic closure of K (necessarily separable because K is perfect). The field extension Ks/K is infinite, and the Galois group is accordingly given the Krull topology. The group Z ^ {\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 Kn of degree n for each integer n ≥ 1, and that the union of these extensions is equal to Ks. Moreover, as part of the structure of the quasi-finite field, there is a generator Fn for each Gal(Kn/K), and the generators must be coherent, in the sense that if n divides m, the restriction of Fm to Kn is equal to Fn.
Examples The most basic example, which motivates the definition, is the finite field K = Fq. It has a unique cyclic extension of degree n, namely Kn = Fqn. The union of the Kn is the algebraic closure Ks. We take Fn to be the Frobenius element; that is, Fn(x) = xq. Another example is K = C((T)), the ring of formal Laurent series in T over the field C of complex numbers. (These are simply formal power series in which we also allow finitely many terms of negative degree.) Then K has a unique cyclic extension
K n = C ( ( T 1 / n ) ) {\displaystyle K_{n}=\mathbf {C} ((T^{1/n}))}
of degree n for each n ≥ 1, whose union is an algebraic closure of K called the field of Puiseux series, and a generator of Gal(Kn/K) is given by
F n ( T 1 / n ) = e 2 π i / n T 1 / n . {\displaystyle F_{n}(T^{1/n})=e^{2\pi i/n}T^{1/n}.}
This construction works if C is replaced by any algebraically closed field C of characteristic zero.
See also Pseudo-finite field
Notes
References Artin, Emil; Tate, John (2009) [1967], Class field theory, American Mathematical Society, ISBN 978-0-8218-4426-7, MR 2467155, Zbl 1179.11040 Serre, Jean-Pierre (1979), Local Fields, Graduate Texts in Mathematics, vol. 67, translated by Greenberg, Marvin Jay, Springer-Verlag, ISBN 0-387-90424-7, MR 0554237, Zbl 0423.12016
