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 {\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 extension K_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.

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