Quasi-finite field

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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.¹

1 Formal definition
2 Examples
3 Notes
4 References

[edit] Formal definition

A quasi-finite field is a perfect field K together with an isomorphism of topological groups

$\phi : \hat\mathbf Z \to \operatorname{Gal}(K_s/K),$

where K_s is an algebraic closure of K (necessarily separable because K is perfect). 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.

[edit] 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.) It can be shown that K has a unique cyclic extension

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