Quasi-finite field

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

[math]\displaystyle{ \phi : \hat{\mathbf Z} \to \operatorname{Gal}(K_s/K), }[/math]

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 [math]\displaystyle{ \widehat{\mathbf{Z}} }[/math] 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

[math]\displaystyle{ K_n = \mathbf C((T^{1/n})) }[/math]

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 by

[math]\displaystyle{ F_n(T^{1/n}) = e^{2\pi i/n} T^{1/n}. }[/math]

This construction works if C is replaced by any algebraically closed field C of characteristic zero.

References

• Artin, Emil; Tate, John (2009) [1967], Class field theory, American Mathematical Society, ISBN 978-0-8218-4426-7 
• Serre, Jean-Pierre (1979), Local Fields, Graduate Texts in Mathematics, 67, Springer-Verlag, ISBN 0-387-90424-7