Quasi-finite field

In mathematics, a quasi-finite field[1] 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.[2]

Formal definition

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

ϕ : Z ^ → Gal ⁡ ( K s / K ) , {\displaystyle \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 Z ^ {\displaystyle {\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.[3] 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 = C ( ( T 1 / n ) ) {\displaystyle 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 π i / n T 1 / n . {\displaystyle 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.[4]

Notes

  1. ^ (Artin & Tate 2009, §XI.3) say that the field satisfies "Moriya's axiom"
  2. ^ As shown by Mikao Moriya (Serre 1979, chapter XIII, p. 188)
  3. ^ (Serre 1979, §XIII.2 exercise 1, p. 192)
  4. ^ (Serre 1979, §XIII.2, p. 191)

References