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 where "K" is an algebraic closure of "K" (necessarily separable because "K" is perfect). The field extension "K"/"K" is infinite, and the Galois group is accordingly given the Krull topology . The group formula_2 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" of degree "n" for each integer "n" ≥ 1, and that the union of these extensions is equal to "K". Moreover, as part of the structure of the quasi-finite field, there is a generator "F" for each Gal("K"/"K"), and the generators must be "coherent", in the sense that if "n" divides "m", the restriction of "F" to "K" is equal to "F". 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" = GF("q"). The union of the "K" is the algebraic closure "K". We take "F" to be the Frobenius element; that is, "F"("x") = "x". 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 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"/"K") is given by This construction works if C is replaced by any algebraically closed field "C" of characteristic zero.