# 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

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 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}.^{[3]} 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

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

This construction works if **C** is replaced by any algebraically closed field *C* of characteristic zero.^{[4]}

## Notes

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

## References

- Artin, Emil; Tate, John (2009) [1967],
*Class field theory*, American Mathematical Society, ISBN 978-0-8218-4426-7, MR 2467155, Zbl 1179.11040 - Serre, Jean-Pierre (1979),
*Local fields*, Graduate Texts in Mathematics**67**, Translated from the French by Marvin Jay Greenberg, Springer-Verlag, ISBN 0-387-90424-7, MR 554237, Zbl 0423.12016