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Mathematical terminology
a Galois module is a G-module, with G being the Galois group (named for Évariste Galois) of some extension of fields. The term Galois representation is
Galois_representation
Mathematical arithmetic dynamics function
arithmetic dynamics, an arboreal Galois representation is a continuous group homomorphism between the absolute Galois group of a field and the automorphism
Arboreal Galois representation
Arboreal_Galois_representation
Mathematical group
In Galois theory, a branch of abstract algebra, the Galois group of a certain type of field extension is a symmetry group characterizing how it extends
Galois_group
1995 publication in mathematics
about Galois representations of elliptic curves. He then uses this result to prove that all semistable curves are modular, by proving that the Galois representations
Wiles's proof of Fermat's Last Theorem
Wiles's_proof_of_Fermat's_Last_Theorem
Result concerning properties of Galois representations associated with modular forms
true. In mathematical terms, Ribet's theorem shows that if the Galois representation associated with an elliptic curve has certain properties, then that
Ribet's_theorem
Authenticated encryption mode for block ciphers
encryption and uses arithmetic in the Galois field GF(2128) to compute the authentication tag, hence its name. Galois Message Authentication Code (GMAC)
Galois/Counter_Mode
Branch of mathematics that studies abstract algebraic structures
Asymptotic representation theory, Lecture notes 2009–2010 https://ncatlab.org/nlab/show/asymptotic+representation+theory Galois representation Glossary
Representation_theory
Galois deformation Galois descent Galois extension Galois field Galois geometry Galois group Absolute Galois group Galois LFSRs Galois module Galois representation
List of things named after Évariste Galois
List_of_things_named_after_Évariste_Galois
Conjecture in algebraic geometry
cycles on a variety in terms of a more computable invariant, the Galois representation on étale cohomology. The conjecture is a central problem in the
Tate_conjecture
Linear representation in mathematics
mathematics, the Steinberg representation, or Steinberg module or Steinberg character, denoted by St, is a particular linear representation of a reductive algebraic
Steinberg_representation
Conjecture in number theory
Serre (1975, 1987), states that an odd, irreducible, two-dimensional Galois representation over a finite field arises from a modular form. A stronger version
Serre's_modularity_conjecture
named after Bernard Dwork, is the L-function attached to the p-adic Galois representation arising from the p-adic etale cohomology of an algebraic variety
Dwork_conjecture
Algebraic structure
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that has a finite number of elements. As with any field
Finite_field
deformation ring is a ring that controls liftings of a representation of a profinite group (usually a Galois group) from a finite field to a local ring. In particular
Deformation_ring
Mathematical theory
a p {\displaystyle p} -adic representation of K {\displaystyle K} (or of G K {\displaystyle G_{K}} , the absolute Galois group of K {\displaystyle K}
P-adic_Hodge_theory
Monoidal category
groups Mumford–Tate group and motivic Galois group arise from categories of Hodge structures, category of Galois representations and motives through Tannakian
Tannakian_formalism
Algebraic structure
over Zp with a linear action of the absolute Galois group GK of K. Thus, it is a Galois representation also referred to as the p-adic cyclotomic character
Tate_module
Describes statistically the splitting of primes in a given Galois extension of Q
has degree 4 and is abelian, with the Galois group isomorphic to the Klein four-group. It turned out that the Galois group of the extension plays a key role
Chebotarev_density_theorem
Conjectures connecting number theory and geometry
relate the structure of Galois groups in algebraic number theory to automorphic forms and, more generally, the representation theory of algebraic groups
Langlands_program
integral ideal, which is analogous to the Artin conductor of a Galois representation. It is given as a product of prime ideals, together with associated
Conductor of an elliptic curve
Conductor_of_an_elliptic_curve
Group comohology of Galois modules
mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups
Galois_cohomology
Type of finite commutative rings
Galois rings are a type of finite commutative rings which generalize both the finite fields and the rings of integers modulo a prime power. A Galois ring
Galois_ring
Branch of mathematics
of deformation theory is with Galois deformations. It allows us to answer the question: If we have a Galois representation G → GL n ( F p ) {\displaystyle
Deformation_(mathematics)
Suppose that L {\displaystyle L} is a finite Galois extension of the local field K {\displaystyle K} , with Galois group G {\displaystyle G} . If χ {\displaystyle
Artin_conductor
associated Galois representations have finite image. The dihedral case is the easiest case of the theorem because the Galois representation is induced
Langlands–Tunnell_theorem
p^{-1}(xg)} is a well-defined linear map. Galois Galois representation. good A good filtration of a representation of a reductive group G is a filtration
Glossary of representation theory
Glossary_of_representation_theory
French mathematician (1944–2019)
81, 1985, p. 515). He introduced the concept of geometric Galois representation of the Galois group of a number field. He also worked on Bloch-Kato conjectures
Jean-Marc_Fontaine
Mathematics concept
question of the Galois representation on the Tate module of an abelian variety A. Conjecturally, the image of such a Galois representation, which is an l-adic
Mumford–Tate_group
Arithmetic in a field with a finite number of elements
GF(pn) and is also called the Galois field of order pn, in honor of the founder of finite field theory, Évariste Galois. GF(p), where p is a prime number
Finite_field_arithmetic
American mathematician (born 1944)
1980s, he introduced the notion of a Selmer group for a p-adic Galois representation and generalized the "main conjectures" of Iwasawa and Barry Mazur
Ralph_Greenberg
Algebraic structure with addition, multiplication, and division
are central to differential Galois theory, a variant of Galois theory dealing with linear differential equations. Galois theory studies algebraic extensions
Field_(mathematics)
Completes the Langlands program for general linear groups over algebraic function fields
generalizing class field theory of function fields from abelian Galois groups to non-abelian Galois groups. The Langlands conjectures for GL1(K) follow from
Lafforgue's_theorem
American mathematician (born 1933)
concept of projective modules, and for the Swan representation, an l-adic projective representation of a Galois group. His work has mainly been in the area
Richard_Swan
French mathematician
bundles on curves and p-adic Hodge theory, in: Automorphic Forms and Galois Representation, London Mathematical Society Lecture Note Series, Volume 415, Cambridge
Laurent_Fargues
Mathematical function associated to algebraic varieties
good reduction, in a definite sense, at all primes p for which the Galois representation ρ on the étale cohomology groups of V is unramified. For those,
Hasse–Weil_zeta_function
American mathematician (1862–1932)
1893 the classification of the structure of finite fields (also called Galois fields). Around 1900, he began working on the foundations of geometry. He
E._H._Moore
Mathematical operation on Galois modules
an operation on Galois modules. For example, if K is a field, GK is its absolute Galois group, and ρ : GK → AutQp(V) is a representation of GK on a finite-dimensional
Tate_twist
Branch of mathematics that studies the properties of groups
equations of high degree. Évariste Galois coined the term "group" and established a connection, now known as Galois theory, between the nascent theory
Group_theory
Conjecture". Galois theory of p-extensions. Springer Science & Business Media. p. 180. ISBN 9783662049679. Calegari, Frank (2011). "Even Galois representations
Fontaine–Mazur_conjecture
Multivariate functions can be written using univariate functions and summing
that only contain radicals and arithmetic operations. For higher orders, Galois theory shows us that the solutions of algebraic equations cannot be expressed
Kolmogorov–Arnold representation theorem
Kolmogorov–Arnold_representation_theorem
Representation theory of groups
integer combination. The reasons are studied in depth in Galois module theory. The regular representation of a group ring is such that the left-hand and right-hand
Regular_representation
Type of group in abstract algebra
group on a set of size n is the Galois group of the general polynomial of degree n and plays an important role in Galois theory. In invariant theory, the
Symmetric_group
Set with associative invertible operation
Évariste Galois, in the 1830s, introduced the term group (French: groupe) for the symmetry group of the roots of an equation, now called a Galois group.
Group_(mathematics)
Field of mathematics
Arithmetic topology Combinatorics and dynamical systems Arboreal Galois representation Silverman, Joseph H. (2007). The Arithmetic of Dynamical Systems
Arithmetic_dynamics
of a Galois group giving the Galois action on a group of roots of unity. As a one-dimensional representation over a ring R, its representation space
Cyclotomic_character
Mathematical group that can be generated as the set of powers of a single element
prime, then Z/pZ is a finite field, and is usually denoted Fp or GF(p) for Galois field. For every positive integer n, the set of the integers modulo n that
Cyclic_group
Type of generalization of periodic functions in Euclidean space
constructs automorphic forms and their correspondent functions as embeddings of Galois groups to their underlying global field extensions. In this formulation
Automorphic_form
{\displaystyle B\mathbb {G} _{m}} are 1 (resp. 0), and the l-adic Galois representation on the (2n)th cohomology group is the nth power of the cyclotomic
Behrend's_trace_formula
Canadian mathematician
The University of Western Ontario. His research interests include Galois groups, Galois cohomology, quadratic forms, and nonlinear dynamics. Mináč received
Ján_Mináč
Number with a real and an imaginary part
theorem, or topological ones such as the winding number, or a proof combining Galois theory and the fact that any real polynomial of odd degree has at least
Complex_number
Structure in algebraic geometry
motivic Galois group has the surrounding representation theory. (What it is not, is a Galois group; however in terms of the Tate conjecture and Galois representations
Motive_(algebraic_geometry)
Chinese-American mathematician (born 1982)
S2CID 5317997. Yun, Zhiwei (2014). "Motives with exceptional Galois groups and the inverse Galois problem". Inventiones Mathematicae. 196 (2): 267–337. arXiv:1112
Zhiwei_Yun
Mathematical behavior near singularities
This extension is generally not Galois but has Galois closure L ( f ) {\displaystyle L(f)} . The associated Galois group of the extension [ L ( f ) :
Monodromy
is a special kind of basis for Galois extensions of finite degree, characterised as forming a single orbit for the Galois group. The normal basis theorem
Normal_basis
smaller set of the visible ones. The real and the abstract state spaces are Galois connected. This means that if we take an element from the abstract space
Abstract_model_checking
Added a basic definition in group theory and algebra
(1984) and named by Matzat (1987). It appears in Galois theory, in the study of the inverse Galois problem or the embedding problem which is a generalization
Semiabelian_group
Type of Dirichlet series associated to number field extensions
{\displaystyle L/K} H {\displaystyle H} be Galois group for M / L {\displaystyle M/L} G {\displaystyle G} be Galois group for M / K {\displaystyle M/K} ⟨ ⋅
Artin_L-function
Partially ordered set in which all subsets have both a supremum and infimum
characterized as the lower adjoint part of a unique Galois connection. For any pair of preorders X and Y, a Galois connection is given by a pair of monotone functions
Complete_lattice
Formal language used to construct ontologies
aggregates and inductive definitions) KIF Domain theory Formal concept analysis Galois connection Lattice (order) Modeling language OntoUML Kuhn, Tobias. "Attempto
Ontology_language
Mathematical abelian group
subgroups of S4. According to Galois theory, the existence of the Klein four-group (and in particular, the permutation representation of it) explains the existence
Klein_four-group
Concept in mathematics
index of G consists of the root datum of Gksep, the Galois action on its Dynkin diagram, and a Galois-invariant subset of the vertices of the Dynkin diagram
Reductive_group
German mathematician (1882–1935)
subgroups of the Galois group. In 1918, Noether published a paper on the inverse Galois problem. Instead of determining the Galois group of transformations
Emmy_Noether
Number with an integer power equal to 1
of integers modulo n and the Galois group of Q ( ω ) . {\displaystyle \mathbb {Q} (\omega ).} This shows that this Galois group is abelian, and implies
Root_of_unity
Canadian mathematician
web of conjectures and results connecting representation theory and automorphic forms to the study of Galois groups in number theory, for which he received
Robert_Langlands
Jean-Marc Fontaine that are used to classify p {\displaystyle p} -adic Galois representations. The ring B d R {\displaystyle \mathbf {B} _{dR}} is defined
Fontaine's_period_rings
Group controlling representation theory
group that controls the representation theory of G. If G is defined over a field k, then LG is an extension of the absolute Galois group of k by a complex
Langlands_dual_group
German mathematician
theory, the asymptotic distribution of Galois groups of number fields, and with the inverse problem of Galois theory. In 1993 he began a collaboration
Gunter_Malle
Topics referred to by the same term
line or vector that is perpendicular to a given object Normal basis (of a Galois extension), used heavily in cryptography Normal bundle Normal cone, of a
Normal
theory to define important subcategories of p-adic Galois representations of the absolute Galois group of local and global fields. Let G be a group and
B-admissible_representation
German mathematician (1849–1917)
Galois group is p mod m. From this point of view, the distribution of Frobenius conjugacy classes in Galois groups over Q (or, more generally, Galois
Ferdinand_Georg_Frobenius
Sporadic simple group
The inverse Galois problem seems to be unsolved for M23. In other words, no polynomial in Z[x] seems to be known to have M23 as its Galois group. The inverse
Mathieu_group_M23
Class of differential equations expressible in differential algebra
hypergeometric equation. In differential Galois theory the case of algebraic solutions is that in which the differential Galois group G is finite (equivalently
Algebraic differential equation
Algebraic_differential_equation
History of a branch of mathematics
Évariste Galois is honored as the first mathematician linking group theory and field theory, with the theory that is now called Galois theory. Galois also
History_of_group_theory
Finite extension of the rationals
inertia group measures the difference between the local Galois groups at some place and the Galois groups of the involved finite residue fields. The following
Algebraic_number_field
Russian mathematician (1937–2010)
he was 19. He co-founded three new branches of mathematics: topological Galois theory (with his student Askold Khovanskii), KAM theory (with Andrey Kolmogorov
Vladimir_Arnold
Non-abelian group of order eight
relate the quaternion group to Galois theory. In 1936 Ernst Witt published his approach to the quaternion group through Galois theory. In 1981, Richard Dean
Quaternion_group
General concept and operation in mathematics
such a more general duality is from Galois theory. For a fixed Galois extension K / F, one may associate the Galois group Gal(K/E) to any intermediate
Duality_(mathematics)
Number-theoretic concept
of p-adic integers. This group is important because of its relation to Galois theory, étale homotopy theory, and the ring of adeles. In addition, it provides
Profinite_integer
Spanish mathematician
congruences of degrees modulo p with I. M. Isaacs, and with Galois automorphisms: the Galois-McKay conjecture). Together with I. M. Isaacs and G. Malle
Gabriel_Navarro_Ortega
Permutation group that preserves no non-trivial partition
introduced by Évariste Galois in his last letter, in which he used the French term équation primitive for an equation whose Galois group is primitive. In
Primitive_permutation_group
On the reciprocity law in algebraic number fields
for that Galois group and this representation, then Langlands reciprocity conjecture says that there exists automorphic cuspidal representation π {\displaystyle
Hilbert's_ninth_problem
Duality for Galois modules for the absolute Galois group of a non-archimedean local field
In Galois cohomology, local Tate duality (or simply local duality) is a duality for Galois modules for the absolute Galois group of a non-archimedean
Local_Tate_duality
Topological group with compact topology
from it. In fact any profinite group is a compact group. This means that Galois groups are compact groups, a basic fact for the theory of algebraic extensions
Compact_group
Construction in group theory
constructed by Évariste Galois in the 1830s, and were the second family of finite simple groups, after the alternating groups. Galois constructed them as
Projective_linear_group
Branch of mathematics that studies algebraic structures
Symmetric function Formally real field Real closed field Galois theory Galois group Inverse Galois problem Kummer theory Module (mathematics) Bimodule Annihilator
List of abstract algebra topics
List_of_abstract_algebra_topics
Representation of a computer program
property graph concept that models details of cloud deployments. Galois' CPG for LLVM. Galois Inc. provides a code property graph based on the LLVM compiler
Code_property_graph
Group that is also a differentiable manifold with group operations that are smooth
differential equations, in much the same way that finite groups are used in Galois theory to model the discrete symmetries of algebraic equations. Sophus Lie
Lie_group
Index of articles associated with the same name
related to its inverse Adjoint equation The upper and lower adjoints of a Galois connection in order theory The adjoint of a differential operator with general
Adjoint
Mathematical conjectures in class field theory
generalization of local class field theory from abelian Galois groups to non-abelian Galois groups. The local Langlands conjectures for GL 1 ( K )
Local_Langlands_conjectures
Method of deriving an ontology
Galois connection between sets of objects and of attributes. This is why in French a concept lattice is sometimes called a treillis de Galois (Galois
Formal_concept_analysis
British mathematician
mathematician, famous for his major results and conjectures on Galois module theory in the Galois structure of rings of integers. He was born in Munich to a
Albrecht_Fröhlich
Mathematical version of an order change
This line of work ultimately resulted, through the work of Évariste Galois, in Galois theory, which gives a complete description of what is possible and
Permutation
Mathematical function
"character in nLab". ncatlab.org. Retrieved 2017-10-31. Artin, Emil (1966), Galois Theory, Notre Dame Mathematical Lectures, number 2, Arthur Norton Milgram
Character_(mathematics)
Negative integer two units from the origin in mathematics
Emmrich and Clark Lyons (December 18, 2017). "Norm-Euclidean Ideals in Galois Cubic Fields" (PDF). 2017 West Coast Number Theory Conference. Archived
−2
results of Tate (1967) on p-divisible groups. Suppose that G is the absolute Galois group of a p-adic field K. Then G has a canonical cyclotomic character χ
Hodge–Tate_module
a finite field Fq. Local class field theory gives a description of the Galois group G of the maximal abelian extension of a local field K via the reciprocity
Local_class_field_theory
French mathematician (born 1962)
functor") from representation of G L 2 ( Q p ) {\displaystyle \mathrm {GL} _{2}(\mathbb {Q} _{p})} to representations of the absolute Galois group of Q p
Pierre_Colmez
Polynomial invariant under variable permutations
details. Symmetric polynomials are important to linear algebra, representation theory, and Galois theory. They are also important in combinatorics, where they
Symmetric_polynomial
Branch of mathematics
finite fields. Galois theory explores the relation between field theory and group theory, relying on the fundamental theorem of Galois theory. Besides
Algebra
extensions of the Galois theory in categories and variable categories, or indexed/'parametrized' categories. The Joyal–Tierney representation theorem for topoi
Nonabelian_algebraic_topology
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