Quasi-finite field explained

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 field K together with an isomorphism of topological groups

\phi : \hat{ Z} \to \operatorname{Gal}(K _{s/K),} where K_s is an algebraic closure of K (necessarily separable because K is perfect). The field extension K_s/K is infinite, and the Galois group is accordingly given the Krull topology. The group \widehat{ 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. 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.

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} )) of degree n for each n ≥ 1, whose union is an algebraic closure of K called the field of Puiseux series, and that a generator of Gal(K_n/K) is given byF ^{1 /n} _{n(T} ) = e ^{2\pi} T ^{1/n} . This construction works if C is replaced by any algebraically closed field C of characteristic zero.

Notes and References

1. say that the field satisfies "Moriya's axiom"
2. As shown by Mikao Moriya