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 field K together with an isomorphism of topological groups

${\displaystyle \phi$ :{\hat {\mathbf {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 ${\displaystyle {\widehat {\mathbf {Z} }}}$ 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.

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

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

of degree n for each n ≥ 1, whose union is an algebraic closure of K called the field of Puiseux series, and that a generator of Gal(K_n/K) is given by

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

This construction works if C is replaced by any algebraically closed field C of characteristic zero.[4]

Notes

1. (Artin & Tate 2009, §XI.3) say that the field satisfies "Moriya's axiom"
2. As shown by Mikao Moriya (Serre 1979, chapter XIII, p. 188)
3. (Serre 1979, §XIII.2 exercise 1, p. 192)
4. (Serre 1979, §XIII.2, p. 191)

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, 67, translated by Greenberg, Marvin Jay, Springer-Verlag, ISBN 0-387-90424-7, MR 0554237, Zbl 0423.12016