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Locally compact infinite topological field
Non-Archimedean local fields can also be defined as those fields which are complete with respect to a metric induced by a discrete valuation whose residue field is
Local_field
Book by Jean-Pierre Serre
into English as Local Fields by Marvin Jay Greenberg in 1979, is a seminal graduate-level algebraic number theory text covering local fields, ramification
Local_Fields
Transient electrical signals
Local field potentials (LFP) are transient electrical signals generated in nerves and other tissues by the summed and synchronous electrical activity
Local_field_potential
Mathematical topic
finite residue field. Let L / K {\displaystyle L/K} be a finite Galois extension of nonarchimedean local fields with finite residue fields ℓ / k {\displaystyle
Finite extensions of local fields
Finite_extensions_of_local_fields
Discrete valuation field
multi-dimensional local fields. On the usual local fields (typically completions of number fields or the quotient fields of local rings of algebraic curves) there
Higher_local_field
Algebraic structure with addition, multiplication, and division
known fields are the field of rational numbers, the field of real numbers, and the field of complex numbers. Many other fields, such as fields of rational
Field_(mathematics)
local class field theory (LCFT), introduced by Helmut Hasse, is the study of abelian extensions of local fields; here, "local field" means a field which
Local_class_field_theory
Conjectures connecting number theory and geometry
groups over local fields (with different subcases corresponding to archimedean local fields, p-adic local fields, and completions of function fields) Automorphic
Langlands_program
Mathematical concept
In mathematics, a global field is one of two types of fields (the other one is local fields) that are characterized using valuations, or absolute values
Global_field
Algebraic variety with a group structure
explicit in some cases, such as over the real or p-adic fields, and thereby over number fields via local-global principles. Abelian varieties are connected
Algebraic_group
algebraic number theory, the conductor of a finite abelian extension of local or global fields provides a quantitative measure of the ramification in the extension
Conductor (class field theory)
Conductor_(class_field_theory)
Branching out of a mathematical structure
extensions of a valuation of a field K to an extension field of K. This generalizes the notions in algebraic number theory, local fields, and Dedekind domains
Ramification_(mathematics)
algebraic number fields in the p-adic context. With the discrete topology, all fields are locally compact. A locally compact field that is not discrete
Locally_compact_field
Mathematics award
awarded the Fields Medal as of 2026[update]. The most recent group of Fields Medalists received their awards on 23 July 2026, in Philadelphia. Fields was instrumental
Fields_Medal
the computations of K-theory of global fields (such as number fields and function fields), as well as local fields (such as p-adic numbers). Suslin (1983)
K-groups_of_a_field
Lusztig (1995) classified the unipotent characters over non-archimedean local fields. Vogan (1987) discusses several different possible definitions of unipotent
Unipotent_representation
Function used in local class field theory related to reciprocity laws
(–, –) from K× × K× to the group of nth roots of unity in a local field K such as the fields of reals or p-adic numbers. It is related to reciprocity laws
Hilbert_symbol
Mathematical conjectures in class field theory
linear groups over local fields. The local Langlands conjecture for GL 2 {\displaystyle \operatorname {GL} _{2}} of a local field says that there is a
Local_Langlands_conjectures
Branch of algebraic number theory concerned with abelian extensions
fields are not extendable to the general case of algebraic number fields, and different conceptual principles are in use in the general class field theory
Class_field_theory
Mathematical property of algebraic structures
theory of rank one valued fields and normed spaces over rank one valued fields as follows. Let K {\displaystyle K} be a field endowed with an absolute
Archimedean_property
American football player (born 2003)
year against Pittsburgh, Fields caught his first career touchdown in addition to five receptions for 58 yards. As a junior, Fields emerged as one of the
Malachi_Fields
notions is motivated by the local–global principle that relates properties of a number field with properties of all its local fields. The definition of an order
Order_(ring_theory)
Finite extension of the rationals
at a local level first, that is to say, by looking at the corresponding local fields. For number fields K {\displaystyle K} , the local fields are the
Algebraic_number_field
Algebra term
their places such that the corresponding local fields are Witt equivalent. In particular, two number fields K and L are Witt equivalent if and only if
Witt_group
Tool for solving polynomial equations
over local fields, or more generally, over ultrametric fields. In the original case, the ultrametric field of interest was essentially the field of formal
Newton_polygon
Mathematical group
are defined and have certain standardized properties. Fields can be extended into larger fields with the same operations, such as how Q {\displaystyle
Galois_group
Used to compare mixed characteristic situations with purely finite characteristic ones
such as local fields of characteristic zero which have residue fields of characteristic prime p. A perfectoid field is a complete topological field K whose
Perfectoid_space
Concept in class field theory
topology is "locally profinite".) For global fields of characteristic p > 0 {\displaystyle p>0} (function fields), the Weil group is the subgroup of the absolute
Weil_group
Topics referred to by the same term
Look up Elysian Fields in Wiktionary, the free dictionary. The Elysian Fields, also called Elysium, are the final resting place of the souls of the heroic
Elysian_Fields
finite field is quasi-algebraically closed by the Chevalley–Warning theorem. Algebraic function fields of dimension 1 over algebraically closed fields are
Quasi-algebraically closed field
Quasi-algebraically_closed_field
Map raising elements to the pth power, in characteristic p
local fields, there is a concept of Frobenius endomorphism that induces the Frobenius endomorphism in the corresponding extension of residue fields.
Frobenius_endomorphism
2011 American crime film by Ami Canaan Mann
Texas Killing Fields (also known as The Fields) is a 2011 American crime film directed by Ami Canaan Mann and starring Sam Worthington, Jeffrey Dean Morgan
Texas_Killing_Fields_(film)
Filtration of the Galois group of a local field extension
more specifically in local class field theory, the ramification groups are a filtration of the Galois group of a local field extension, which gives
Ramification_group
Branch of number theory
algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring
Algebraic_number_theory
Russian mathematician
symbol on local fields and higher local field, higher class field theory, p-class field theory, arithmetic noncommutative local class field theory. He
Ivan_Fesenko
algebras over a field. The concept is named after Helmut Hasse. The invariant plays a role in local class field theory. Let K be a local field with valuation
Hasse_invariant_of_an_algebra
Theorem in algebraic number theory
Galois extension of local or global fields to an extension of Galois groups. It was introduced by Shafarevich (1946) for local fields and by Weil (1951)
Shafarevich–Weil_theorem
American white supremacist (born 1932)
Fields (born September 30, 1932) is an American white supremacist activist and newspaper publisher. A fascist activist from his teenage years, Fields
Ed_Fields
Mathematical formal group law
compositum of all fields formed by adjoining such modules to the original field gives the ramified part. A Lubin–Tate extension of a local field K is an abelian
Lubin–Tate_formal_group_law
Mathematical object
{\text{L}}(k))} . This was proven in over non-Archimedean local fields and later in for all local fields of characteristic zero. For more details on this question
Gelfand_pair
Russian mathematician (1945–2025)
explicit formula to higher local fields is called the Vostokov symbol. It plays an important role in higher local class field theory.[failed verification]
Sergei_Vostokov
American mathematician (1937–2006)
finite fields". Annals of Mathematics. Series 2. 88 (2): 239–271. doi:10.2307/1970573. JSTOR 1970573. Ax, James (1970). "Zeros of polynomials over local fields—The
James_Ax
Linear representation in mathematics
(sometimes called special representations) for algebraic groups over local fields. For the general linear group GL(2), the dimension of the Jacquet module
Steinberg_representation
Mathematical theorem
Springer-Verlag, ISBN 3-540-65399-6, Zbl 0956.11021 Serre, Jean-Pierre (1979), Local Fields, Graduate Texts in Mathematics, vol. 67, translated by Greenberg, Marvin
Artin_reciprocity
German mathematician
Michael Rapoport—proved the local Langlands conjectures for the general linear group GLn(K) for positive characteristic local fields K. An alumnus of the University
Ulrich_Stuhler
Suburb of Sydney, New South Wales, Australia
Nirimba Fields is a suburb of Sydney in the state of New South Wales, Australia. Nirimba Fields is in north-west Sydney in the local government area of
Nirimba_Fields
Caldera volcano west of Naples, Italy
The Phlegraean Fields is monitored by the Vesuvius Observatory. Part of the city of Naples is built over it. The Phlegraean Fields' largest known eruptions
Phlegraean_Fields
Hilbert symbol of a local field. The name "explicit reciprocity law" refers to the fact that the Hilbert symbols of local fields appear in Hilbert's reciprocity
Explicit_reciprocity_law
Abelian group related to division algebras
results are only for special classes of fields. For example, if K {\displaystyle K} is a global field or local field, then a division algebra of any degree
Brauer_group
quasi-finite field is a generalisation of a finite field. Standard local class field theory usually deals with complete valued fields whose residue field is finite
Quasi-finite_field
over the finite fields, local fields nor global function fields. Image Special fields This type exists over any finite field, any local non-archimedean
List of irreducible Tits indices
List_of_irreducible_Tits_indices
Algebraic structure
characteristic zero and all finite fields are perfect. Perfect fields are significant because Galois theory over these fields becomes simpler, since the general
Perfect_field
Japanese mathematician
2011) was a Japanese mathematician who worked on algebraic groups over local fields who introduced Iwahori–Hecke algebras and Iwahori subgroups. Iwahori
Nagayoshi_Iwahori
Prize awarded by the American Mathematical Society
James; Kochen, Simon (1966). "Diophantine problems over local fields III. Decidable fields". Annals of Mathematics. 83 (3): 437–456. doi:10.2307/1970476
Cole_Prize
defined for a finite degree extension of local fields. It plays a basic role in Pontryagin duality for p-adic fields. The relative different δL / K is defined
Different_ideal
Concept in ring theory
when extended to the algebraic closure of its base field Serre, Jean-Pierre. (1979). Local Fields. New York, NY: Springer New York. ISBN 978-1-4757-5673-9
Azumaya_algebra
Allows one to kill tame ramification by taking an extension of a base field
extension of A. Finite extensions of local fields Cornell, Gary (1982), "On the Construction of Relative Genus Fields", Transactions of the American Mathematical
Abhyankar's_lemma
Concept in number theory
combines all local versions of a global field into one object. For the rational numbers, these local versions include the real numbers and the fields of p {\displaystyle
Adele_ring
Topics referred to by the same term
local symbol used to formulate Weil reciprocity A Steinberg symbol on a local field This disambiguation page lists mathematics articles associated with the
Local_symbol
Suburb of Sydney, New South Wales, Australia
Macquarie Fields is a suburb of Sydney, in the state of New South Wales, Australia. Macquarie Fields is located 38 kilometres south-west of the Sydney
Macquarie_Fields
American comedian, actor, juggler and writer (1880–1946)
personal notes in grandson Ronald Fields's book W. C. Fields by Himself, it was shown that Fields was married (and subsequently estranged from his wife)
W._C._Fields
Mathematical theory
classify and study p-adic Galois representations of characteristic 0 local fields with residual characteristic p (such as Qp). The theory has its beginnings
P-adic_Hodge_theory
obstruction is non-trivial, then X may have points over all local fields but not over the global field. The Manin obstruction is sometimes called the Brauer–Manin
Manin_obstruction
Topics referred to by the same term
mathematics, local duality may refer to: Local Tate duality of modules over a Galois group of a local field Grothendieck local duality of modules over local rings
Local_duality
American theoretical physicist (1918–1994)
for much of modern quantum field theory, including a variational approach, and the equations of motion for quantum fields. He developed the first electroweak
Julian_Schwinger
American jazz saxophonist
for a local musician. He is survived by his widow Constance Fields, son Michael Fields, daughter Jacqueline Fields, granddaughter Danielle Fields and great-grandson
Mickey_Fields
Duality for Galois modules for the absolute Galois group of a non-archimedean local field
of tools for computing the Galois cohomology of local fields. Let K be a non-archimedean local field, let Ks denote a separable closure of K, and let
Local_Tate_duality
Urban open space in London, England
current park that opened in 1936. Coram's Fields provides services for children and young people in the local community, including an after school and
Coram's_Fields
Discrete subgroup in a locally compact topological group
groups are semisimple algebraic groups over local fields of characteristic 0, for example the p-adic fields Q p {\displaystyle \mathbb {Q} _{p}} . There
Lattice_(discrete_subgroup)
American mail carrier (c. 1832 – 1914)
research about Mary Fields to the United States Postal Service Archives Historian in 2006. This enabled the USPS to establish Mary Fields' contribution as
Mary_Fields
Public school district in Ohio, U.S.
Districts – District Detail for Field Local". National Center for Education Statistics. Institute of Education Sciences. "Two Fields in One". Akron Beacon Journal
Field_Local_School_District
Computer algebra system
sophisticated computations in algebraic number fields, in global function fields, and in local fields. KASH is the associated command line interface.
KANT_(software)
English writer
Sarah Fields (born 1969) is an English novelist and short story writer, who writes primarily in the crime fiction and thriller genres. Fields is originally
Helen_Fields
Solving integer equations from all modular solutions
when can local solutions be joined to form a global solution? One can ask this for other rings or fields: integers, for instance, or number fields. For number
Hasse_principle
On the existence of zeros of homogeneous polynomials over the p-adic numbers
is a C2 field). Then one shows that if two Henselian valued fields have equivalent valuation groups and residue fields, and the residue fields have characteristic
Ax–Kochen_theorem
Public park in Queens, New York
Flushing Fields is a public park in the northern section of the Flushing neighborhood of Queens in New York City. The site of this park was purchased by
Flushing_Fields
American grocery store in Los Angeles, California, USA
2022-12-31. "About Fields Market". Fields Market. Archived from the original on 2016-03-02. Love, Marianne (December 17, 2024). "Beloved local Fields Market in
Fields_Market
Russian mathematician (1922–1995)
theory of local fields" in 1951 and his doctoral dissertation titled "Regarding the multiplicative groups of normal R-extensions of local fields" in 1967
Zenon_Borevich
abelian extension of nonarchimedean local fields, and NL/K is the field norm. One of the main theorems in local class field theory states that the norm groups
Norm_group
American actor (1898–1975)
early 1950s (television). He was sometimes credited as "Sid Fields" or "Sidney Field". Fields was born Sidney Hirsch Feldman, the son of Hirsch and Mary
Sidney_Fields
fields K that are finitely generated over their prime fields—including as of special interest number fields and finite fields—and over local fields.
Glossary of arithmetic and diophantine geometry
Glossary_of_arithmetic_and_diophantine_geometry
Theory in number theory
the absolute Galois groups of number fields and mixed-characteristic local fields. Section conjecture Class field theory Fiber functor Neukirch–Uchida
Anabelian_geometry
Unincorporated community in the state of Oregon, United States
and restaurant called Fields Station. The 1-mile (1.6 km) radius around that store has below 25 occupants. In 1881, Charles Fields established a homestead
Fields,_Oregon
Theorem in abstract algebra
2009". In 2010, Ngô was awarded the Fields Medal for this proof. Langlands outlined a strategy for proving local and global Langlands conjectures using
Fundamental lemma (Langlands program)
Fundamental_lemma_(Langlands_program)
Recreational parkland in New Jersey, US
The Elysian Fields in Hoboken, New Jersey, was recreational parkland located on the city's northern riverfront in the 19th century. The area was a popular
Elysian Fields (Hoboken, New Jersey)
Elysian_Fields_(Hoboken,_New_Jersey)
Book about number theory
approach handles all 'A-fields' or global fields, meaning finite algebraic extensions of the field of rational numbers and of the field of rational functions
Basic_Number_Theory
Human settlement in England
London Fields is a park in Hackney, London, although the name also refers to the immediate area in Hackney surrounding it and London Fields station. It
London_Fields
Physical theory with fields invariant under the action of local "gauge" Lie groups
corresponding field (usually a vector field) called the gauge field. Gauge fields are included in the Lagrangian to ensure its invariance under the local group
Gauge_theory
British TV sitcom (1984–1986)
U&Drama. In December 2023, repeats of Fresh Fields began airing on That's TV in the UK. Hester and William Fields (McKenzie and Rodgers) are a devoted middle-aged
Fresh_Fields
In mathematics, class field theory is the study of abelian extensions of local and global fields. 1801 Carl Friedrich Gauss proves the law of quadratic
Timeline of class field theory
Timeline_of_class_field_theory
duality conjecture for archimedean local fields was proved by Roger Howe. For p {\displaystyle p} -adic local fields with p {\displaystyle p} odd it was
Theta_correspondence
British TV sitcom (1989–1991)
French Fields is a British television sitcom. It is a sequel/continuation of the series Fresh Fields and ran for 19 episodes from 5 September 1989 to
French_Fields
Whitehead group is trivial for local fields K, and gave examples of fields for which it is not always trivial. For global fields the combined work of several
Kneser–Tits_conjecture
local fields (real numbers, p-adic numbers) are not Hilbertian. The WWA property (weak 'weak approximation', sic) for a variety V over a number field
Thin_set_(Serre)
component of AE (for class field theory), which is trivial for class field theory of non-archimedean local fields and for function fields, but is non-trivial
Class_formation
Kazhdan 1988) is less general but easier to prove. The local trace formula is an analogue over local fields. Jacquet's relative trace formula is a generalization
Arthur–Selberg_trace_formula
American mathematician (1925–2019)
Proceedings of a Conference on Local Fields, Springer-Verlag, pp. 158–183, MR 0231827 Artin, Emil; Tate, John (2009) [1967], Class field theory, AMS Chelsea Publishing
John_Tate_(mathematician)
English actress, singer and comedian (1898–1979)
towns were visited by Fields. A live show of music and entertainment, it was compèred by Fields, who also performed, together with local talents. The tour
Gracie_Fields
American singer
Arabella Fields (née Sarah Arabella Middleton, also known as "Belle Fields, the Black Nightingale"; 31 January 1879 – after 1933) was an African-American
Arabella_Fields
Close surroundings of an atom
experiences the electric fields created by the atom's electrons that surround the nucleus. In addition, the electric fields created by neighbouring atoms
Local_structure
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LOCAL FIELDS
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