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 fields whose residue field is "finite", but the theory applies equally well when the residue field is only assumed quasi-finite.rf|1|Serre188

Formal definition

A quasi-finite field is a perfect field "K" together with an isomorphism of topological groups: $phi\; :\; hatmathbf\; Z\; o\; operatorname\{Gal\}(K\_s/K),$where "K"_"s" is an algebraic closure of "K" (necessarily separable because "K" is perfect). Generally, the field extension "K"_"s"/"K" will be infinite, and the Galois group is accordingly given the Krull topology. The group Z^{^} is the profinite completion of Z with respect to its subgroups of finite index.

This definition amounts to saying that "K" has a unique 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, we must also provide 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("q"^"n"). 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.) It can be shown that "K" has a unique cyclic extension: $K\_n\; =\; mathbf\; C((T^\{1/n\}))$of degree "n" for each "n" ≥ 1, whose union is an algebraic closure of "K", and that a generator of Gal("K"_"n"/"K") is given by: $F\_n(T^\{1/n\})\; =\; e^\{2pi\; i/n\}\; T^\{1/n\}.$In fact, this construction can be generalised to the situation where C is replaced by any algebraically closed field "C" of characteristic zero.rf|2|Serre191

Notes

ent|1| Serre188 Serre p. 188ent|2| Serre191 Serre p. 191

References

*