Login
Login

Quasi-finite field

In mathematics, a quasi-finite field{{harv|Artin|Tate|2009|loc=§XI.3}} say that the field satisfies "Moriya's axiom" 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.As shown by Mikao Moriya {{harv|Serre|1979|loc=chapter XIII, p. 188}}

Formal definition

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

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

where Ks is an algebraic closure of K (necessarily separable because K is perfect). The field extension Ks/K is infinite, and the Galois group is accordingly given the Krull topology. The group \widehat{\mathbb{Z}} 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 Kn of degree n for each integer n ≥ 1, and that the union of these extensions is equal to Ks.{{harv|Serre|1979|loc=§XIII.2 exercise 1, p. 192}} Moreover, as part of the structure of the quasi-finite field, there is a generator Fn for each Gal(Kn/K), and the generators must be coherent, in the sense that if n divides m, the restriction of Fm to Kn is equal to Fn.

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 Kn = GF(qn). The union of the Kn is the algebraic closure Ks. We take Fn to be the Frobenius element; that is, Fn(x) = xq.

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^{1/n}))

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(Kn/K) is given by

: F_n(T^{1/n}) = e^{2\pi i/n} T^{1/n}.

This construction works if C is replaced by any algebraically closed field C of characteristic zero.{{harv|Serre|1979|loc=§XIII.2, p. 191}}

See also

Notes

{{reflist}}

References

  • {{Citation

| last1=Artin

| first1=Emil

| author-link1=Emil Artin

| last2=Tate

| first2=John

| author2-link=John Tate (mathematician)

| title=Class field theory

| publisher=American Mathematical Society

| year=2009

| orig-date=1967

| isbn=978-0-8218-4426-7

| mr=2467155 | zbl=1179.11040

}}

  • {{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 }}

Category:Class field theory

Category:Field (mathematics)