local fields

local fields Local fields are complete non-discrete topological fields with respect to a nontrivial absolute value arising from a discrete valuation. They appear in the study of number theory, algebraic geometry, and representation theory and connect objects like Leopoldt conjecture, Langlands program, Hasse principle, Tate cohomology and Iwasawa theory.

Definition and basic properties

A local field is a field complete for a discrete valuation with finite residue field; typical examples relate to p-adic numbers, finite fields, number fields and their completions. Key structural properties tie to the existence of a valuation ring, a maximal ideal, and a residue field, connecting to results from Kummer theory, Hilbert symbol computations, and propositions appearing in the work of Emil Artin, Helmut Hasse, John Tate, Kurt Hensel and Alexander Grothendieck. Fundamental invariants include the characteristic (equal to 0 or p), the residue characteristic, and the ramification degree appearing in theorems attributed to Paul Rost and developments in local duality.

Classification and examples

Classification splits into characteristic zero and positive characteristic cases, linking to concepts explored by Kurt Hensel and Emil Artin; characteristic zero local fields are finite extensions of the p-adic numbers for primes like those in the study by Jean-Pierre Serre and John Coates, while positive characteristic cases are isomorphic to Laurent series fields over finite fields, central in examples from André Weil and Serre's Local Fields. Concrete instances invoked across literature include completions appearing in the context of Ring of adeles, Eisenstein series, Tate curve constructions, and explicit fields used by David Rohrlich and Haruzo Hida.

Topology and valuation

The topology is induced by a nonarchimedean absolute value tied to a discrete valuation; this connection is foundational in work of Kurt Hensel and plays a role in proofs by Jean-Pierre Serre and John Tate. Ultrametric inequality effects appear in applications to Hasse–Minkowski theorem, Weil conjectures techniques, and to explicit formulas used by Serre and Tate. The valuation gives rise to a valuation ring and maximal ideal with properties exploited in constructions studied by Alexander Grothendieck in the context of étale cohomology and by Pierre Deligne in the study of local monodromy.

Extensions and ramification

Finite extensions of local fields are classified by ramification theory developed by Helmut Hasse, Niels Jacobsen, and others; tame and wild ramification distinctions are central to results by Jean-Pierre Serre, Kazuya Kato, and Serre's Local Fields. Inertia and decomposition groups in Galois extensions play roles comparable to global decomposition at primes appearing in the work of Emil Artin and Ferdinand Frobenius. Wild ramification phenomena connect to deep results by Andrew Wiles and conjectures addressed in the context of the Langlands program, while explicit ramification filtrations are studied in papers by Serre, Kazuya Kato, and Barry Mazur.

Local field arithmetic (rings of integers, units, norms)

The valuation ring (ring of integers), maximal ideal, and unit group structure are central arithmetic invariants used by Iwasawa, Emil Artin, and John Tate. The multiplicative group decomposes into uniformizers and unit subgroups; Hensel's lemma, formulated by Kurt Hensel, provides lifting results relevant to Eisenstein polynomials and Hasse invariants. Norms from extensions, the structure of unit groups (including higher unit groups), and the behaviour of valuations under norms appear throughout work by Iwasawa, Jean-Pierre Serre, Andrew Wiles, and Kazuya Kato and feature in explicit reciprocity laws exploited in proofs by John Coates and Ralph Greenberg.

Galois theory and local class field theory

Local class field theory, developed by Emil Artin, Helmut Hasse, John Tate, and consolidated in expositions by Serre, gives a reciprocity isomorphism between the abelianized Galois group and quotients of the multiplicative group, relating to reciprocity maps used in the Langlands correspondence and in formulations by Robert Langlands. The study of nonabelian extensions links to ramifications of the Langlands program, ramifications studied by Pierre Deligne and Michael Harris, and to explicit descriptions of Galois representations used by Andrew Wiles, Richard Taylor, and Richard Taylor's collaborators. Cohomological tools from Galois cohomology and duality theorems of Tate underpin proofs of reciprocity and local duality, while the local-global interplay connects to results of Hasse, Artin, and applications in the proof of modularity theorems by Wiles and Taylor–Wiles methods.

Category:Number theory