F_p((t))

F_p((t))
Name	F_p((t))
Other names	Field of formal Laurent series over the finite field of p elements
Type	Field, Topological field, Local field (of positive characteristic)
Base field	F_p
Elements	Formal Laurent series
Topology	t-adic (valuation) topology
Completion	Complete with respect to discrete valuation

F_p((t)).

F_p((t)) is the field of formal Laurent series in the indeterminate t over the finite field Galois's field F_p of p elements, introduced as an analogue of Q_p in positive characteristic. It serves as a prototypical local field in algebraic number theory and algebraic geometry, appearing alongside Laurent series constructions and playing a role in the study of curves over finite fields and moduli problems. The field combines algebraic, valuation-theoretic, and topological features studied by mathematicians such as Hasse, Artin, Weil, and Grothendieck.

Definition and basic properties

F_p((t)) consists of formal sums a_n t^n + a_{n+1} t^{n+1} + ... with n ∈ Z and coefficients a_i in F_p; addition and multiplication follow the formal power series rules used by Cauchy and formalized by Weierstrass and Noether. It is a field of characteristic p, has prime subfield F_p, and contains the subring F_pt of formal power series; classical references include works by Emil Artin and Ivan Fesenko. The field is complete with respect to the discrete valuation v_t normalized so that v_t(t)=1, echoing constructions in Hensel's lemma contexts and in analogies with p-adic numbers studied by Kurt Hensel.

Algebraic structure and topology

As a field, F_p((t)) is a one-dimensional vector space over itself and an infinite-dimensional vector space over F_p; linear algebra over this field links to structures studied by Noether and Artin in noncommutative algebra. The ring of integers F_pt is a discrete valuation ring with maximal ideal t F_pt, similar in form to constructions in Dedekind domain theory and to local rings considered by Grothendieck in his work on schemes. Topologically, the t-adic topology makes F_p((t)) a locally compact, nondiscrete topological field when p is finite, an attribute examined by John Tate in harmonic analysis on local fields and used by Serre in local class field theory.

Field extensions and algebraic closure

Finite separable extensions of F_p((t)) correspond to finite extensions of discrete valuation rings via ramification theory developed by Herbrand and Serre; inseparable extensions reflect phenomena particular to characteristic p studied by Artin and Swan. The algebraic closure is not complete, mirroring the situation for Q_p and requiring completion procedures used by Ax and Kochen; the separable closure features a profinite absolute Galois group related to the work of Grothendieck on étale fundamental groups and to Shafarevich's investigations. The structure of extensions interacts with the theory of Drinfeld modules and with moduli problems examined by Deligne and Rapoport.

Valuation and local field aspects

The valuation v_t gives a discrete valuation with value group Z and residue field F_p, placing F_p((t)) among local fields of positive characteristic alongside Laurent series fields over finite fields considered by Iwasawa and Tate. Local duality theorems of Tate and reciprocity laws of Artin apply in adapted forms, while the study of harmonic analysis and representations over F_p((t)) connects to Harish-Chandra and Langlands-style programs in the function field setting explored by Drinfeld and Lafforgue.

Galois theory and ramification

The absolute Galois group of F_p((t)) exhibits wild ramification phenomena characteristic of positive characteristic fields; key contributors to ramification theory include Herbrand, Swan, and Serre. The upper and lower numbering filtrations of Galois groups, studied by Hasse and Arf, classify tame and wild ramification in finite extensions, and the interaction with Artin–Schreier theory and Kummer theory in characteristic p underpins explicit descriptions of cyclic extensions. These tools are central in analogues of local class field theory developed by Tate and refined in geometric settings by Grothendieck.

Applications in number theory and algebraic geometry

F_p((t)) is fundamental in the function field analogue of classical number theory where researchers like Weil, Drinfeld, and Lafforgue translated problems about Riemann hypothesis analogues and automorphic forms to positive characteristic. It appears in the study of algebraic curves over finite fields, local contributions to the Hasse–Weil zeta function, and in the theory of moduli spaces of bundles on curves investigated by Narasimhan and Seshadri. In arithmetic geometry, F_p((t)) provides local models for degeneration problems considered by Grothendieck and Deligne, and it underlies constructions in the geometric Langlands program advanced by Beilinson and Drinfeld.

Examples and computations

Concrete elements include series like t^{-3} + 2 t^{-1} + 0 + ... with coefficients in F_p and operations computed termwise as in the work of Euler and Cauchy on series manipulations. Finite extensions can be presented by adjoining roots of polynomials such as Artin–Schreier polynomials X^p - X - a, a method used by Artin and Schreier to construct cyclic extensions; explicit ramified extensions are analyzed in examples by Serre and Fesenko. Computations of valuations, residue classes, and norm maps are routine in algorithmic arithmetic geometry and in explicit class field constructions influenced by Iwasawa and Tate.

Category:Local fields