- Quasi-finite field
In

mathematics , a**quasi-finite field**is a generalisation of afinite field . Standardlocal 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 aperfect field "K" together with anisomorphism oftopological group s: $phi\; :\; hatmathbf\; Z\; o\; operatorname\{Gal\}(K\_s/K),$where "K"_{"s"}is analgebraic closure of "K" (necessarily separable because "K" is perfect). Generally, thefield extension "K"_{"s"}/"K" will be infinite, and theGalois group is accordingly given theKrull topology . The group**Z**^{^}is theprofinite 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 theFrobenius element ; that is, "F"_{"n"}("x") = "x"^{"q"}.Another example is "K" =

**C**(("T")), the ring offormal Laurent series in "T" over the field**C**ofcomplex number s. (These are simplyformal 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

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