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
: $\phi : \hat \to \operatorname(K_s/K),$
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 $\widehat$ 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''. 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(''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.) Then ''K'' has a unique cyclic extension
: $K_n = \mathbf C((T^))$
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
: $F_n(T^) = e^ T^.$
This construction works if C is replaced by any algebraically closed field ''C'' of characteristic zero.

Notes

References

*
* {{citation , last=Serre , first=Jean-Pierre , authorlink=Jean-Pierre Serre , title= Local Fields , translator-last1=Greenberg, translator-first=Marvin Jay, translator-link1=Marvin Greenberg , series=Graduate Texts in Mathematics , volume=67 , publisher=Springer-Verlag , year=1979 , isbn=0-387-90424-7 , zbl=0423.12016 , mr=554237

Class field theory
Field (mathematics)