Perfectoid space
In mathematics, perfectoid spaces are adic spaces of special kind, which occur in the study of problems of "mixed characteristic", such as local fields of characteristic zero which have residue fields
Tamagawa number
In mathematics, the Tamagawa number of a semisimple algebraic group defined over a global field k is the measure of , where is the adele ring of k. Tamagawa numbers were introduced by Tamagawa, and na
Eisenstein reciprocity
In algebraic number theory Eisenstein's reciprocity law is a reciprocity law that extends the law of quadratic reciprocity and the cubic reciprocity law to residues of higher powers. It is one of the
Quadratic integer
In number theory, quadratic integers are a generalization of the usual integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the for
Hasse invariant of an algebra
In mathematics, the Hasse invariant of an algebra is an invariant attached to a Brauer class of algebras over a field. The concept is named after Helmut Hasse. The invariant plays a role in local clas
Modular forms modulo p
In mathematics, modular forms are particular complex analytic functions on the upper half-plane of interest in complex analysis and number theory. When reduced modulo a prime p, there is an analogous
Ring of integers
In mathematics, the ring of integers of an algebraic number field is the ring of all algebraic integers contained in . An algebraic integer is a root of a monic polynomial with integer coefficients: .
Minkowski space (number field)
In mathematics, specifically the field of algebraic number theory, a Minkowski space is a Euclidean space associated with an algebraic number field. If K is a number field of degree d then there are d
Stephens' constant
Stephens' constant expresses the density of certain subsets of the prime numbers. Let and be two multiplicatively independent integers, that is, except when both and equal zero. Consider the set of pr
Drinfeld module
In mathematics, a Drinfeld module (or elliptic module) is roughly a special kind of module over a ring of functions on a curve over a finite field, generalizing the Carlitz module. Loosely speaking, t
Discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynom
Adelic algebraic group
In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group G over a number field K, and the adele ring A = A(K) of K. It consists of the points of G having
Cyclotomic character
In number theory, a cyclotomic character is a character 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 s
Quadratic reciprocity
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it
Quartic reciprocity
Quartic or biquadratic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x4 ≡ p (mod q) is solvable; the word "reciproc
Ideal norm
In commutative algebra, the norm of an ideal is a generalization of a norm of an element in the field extension. It is particularly important in number theory since it measures the size of an ideal of
Ramification (mathematics)
In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign. The term is also used from the opposite
Global field
In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields:
* Algebraic number field: A
Waldspurger formula
In representation theory of mathematics, the Waldspurger formula relates the special values of two L-functions of two related admissible irreducible representations. Let k be the base field, f be an a
Cubic reciprocity
Cubic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x3 ≡ p (mod q) is solvable; the word "reciprocity" comes from t
Dedekind zeta function
In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK(s), is a generalization of the Riemann zeta function (which is obtained in the case where K is the field
Explicit reciprocity law
In mathematics, an explicit reciprocity law is a formula for the Hilbert symbol of a local field. The name "explicit reciprocity law" refers to the fact that the Hilbert symbols of local fields appear
Totally imaginary number field
In algebraic number theory, a number field is called totally imaginary (or totally complex) if it cannot be embedded in the real numbers. Specific examples include imaginary quadratic fields, cyclotom
Arithmetic and geometric Frobenius
In mathematics, the Frobenius endomorphism is defined in any commutative ring R that has characteristic p, where p is a prime number. Namely, the mapping φ that takes r in R to rp is a ring endomorphi
P-adic valuation
In number theory, the p-adic valuation or p-adic order of an integer n is the exponent of the highest power of the prime number p that divides n.It is denoted .Equivalently, is the exponent to which a
Higher local field
In mathematics, a higher (-dimensional) local field is an important example of a complete discrete valuation field. Such fields are also sometimes called multi-dimensional local fields. On the usual l
Profinite integer
In mathematics, a profinite integer is an element of the ring (sometimes pronounced as zee-hat or zed-hat) where indicates the profinite completion of , the index runs over all prime numbers, and is t
Elementary number
An elementary number is one formalization of the concept of a closed-form number. The elementary numbers form an algebraically closed field containing the roots of arbitrary equations using field oper
Genus character
In number theory, a genus character of a quadratic number field K is a character of the genus group of K. In other words, it is a real character of the narrow class group of K. Reinterpreting this usi
Different ideal
In algebraic number theory, the different ideal (sometimes simply the different) is defined to measure the (possible) lack of duality in the ring of integers of an algebraic number field K, with respe
Lubin–Tate formal group law
In mathematics, the Lubin–Tate formal group law is a formal group law introduced by Lubin and Tate to isolate the local field part of the classical theory of complex multiplication of elliptic functio
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3.
Proofs of quadratic reciprocity
In number theory, the law of quadratic reciprocity, like the Pythagorean theorem, has lent itself to an unusually large number of proofs. Several hundred proofs of the law of quadratic reciprocity hav
Kummer theory
In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction of nth roots of elements of the base field. The theory was ori
Skolem–Mahler–Lech theorem
In additive and algebraic number theory, the Skolem–Mahler–Lech theorem states that if a sequence of numbers satisfies a linear difference equation, then with finitely many exceptions the positions at
Height function
A height function is a function that quantifies the complexity of mathematical objects. In Diophantine geometry, height functions quantify the size of solutions to Diophantine equations and are typica
Abstract analytic number theory
Abstract analytic number theory is a branch of mathematics which takes the ideas and techniques of classical analytic number theory and applies them to a variety of different mathematical fields. The
Abhyankar's lemma
In mathematics, Abhyankar's lemma (named after Shreeram Shankar Abhyankar) allows one to kill tame ramification by taking an extension of a base field. More precisely, Abhyankar's lemma states that if
Conductor-discriminant formula
In mathematics, the conductor-discriminant formula or Führerdiskriminantenproduktformel, introduced by Hasse for abelian extensions and by Artin for Galois extensions, is a formula calculating the rel
Local Euler characteristic formula
In the mathematical field of Galois cohomology, the local Euler characteristic formula is a result due to John Tate that computes the Euler characteristic of the group cohomology of the absolute Galoi
Formal group law
In mathematics, a formal group law is (roughly speaking) a formal power series behaving as if it were the product of a Lie group. They were introduced by S. Bochner. The term formal group sometimes me
Arithmetic dynamics
Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Classically, discrete dynamics refers to the study of the iteration of self-maps of the c
Equivariant L-function
In algebraic number theory, an equivariant Artin L-function is a function associated to a finite Galois extension of global fields created by packaging together the various Artin L-functions associate
Dedekind–Kummer theorem
In algebraic number theory, the Dedekind–Kummer theorem describes how a prime ideal in a Dedekind domain factors over the domain's integral closure.
Carlitz exponential
In mathematics, the Carlitz exponential is a characteristic p analogue to the usual exponential function studied in real and complex analysis. It is used in the definition of the Carlitz module – an e
Integer
An integer is the number zero (0), a positive natural number (1, 2, 3, etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the correspon
Regular prime
In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility
Algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are exp
Brauer group
In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras over K, with addition given by the tensor product of algebras
Heegner number
In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer d such that the imaginary quadratic field has class number 1. Equivalently, its ring of integers has
Hasse principle
In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to pie
Cubic field
In mathematics, specifically the area of algebraic number theory, a cubic field is an algebraic number field of degree three.
Fontaine's period rings
In mathematics, Fontaine's period rings are a collection of commutative rings first defined by Jean-Marc Fontaine that are used to classify p-adic Galois representations.
Newton polygon
In mathematics, the Newton polygon is a tool for understanding the behaviour of polynomials over local fields, or more generally, over ultrametric fields. In the original case, the local field of inte
P-adic Hodge theory
In mathematics, p-adic Hodge theory is a theory that provides a way to classify and study p-adic Galois representations of characteristic 0 local fields with residual characteristic p (such as Qp). Th
Hilbert's ninth problem
Hilbert's ninth problem, from the list of 23 Hilbert's problems (1900), asked to find the most general reciprocity law for the norm residues of k-th order in a general algebraic number field, where k
Stark conjectures
In number theory, the Stark conjectures, introduced by Stark and later expanded by Tate, give conjectural information about the coefficient of the leading term in the Taylor expansion of an Artin L-fu
Totally real number field
In number theory, a number field F is called totally real if for each embedding of F into the complex numbers the image lies inside the real numbers. Equivalent conditions are that F is generated over
Fundamental unit (number theory)
In algebraic number theory, a fundamental unit is a generator (modulo the roots of unity) for the unit group of the ring of integers of a number field, when that group has rank 1 (i.e. when the unit g
Serre's conjecture II (algebra)
In mathematics, Jean-Pierre Serre conjectured the following statement regarding the Galois cohomology of a simply connected semisimple algebraic group. Namely, he conjectured that if G is such a group
Frobenius endomorphism
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic p, an important class whi
Local field
In mathematics, a field K is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation v and if its residue field k is finite. Equivalently, a
Ring class field
In mathematics, a ring class field is the abelian extension of an algebraic number field K associated by class field theory to the ring class group of some order O of the ring of integers of K.
Heegner point
In mathematics, a Heegner point is a point on a modular curve that is the image of a quadratic imaginary point of the upper half-plane. They were defined by Bryan Birch and named after Kurt Heegner, w
Additive polynomial
In mathematics, the additive polynomials are an important topic in classical algebraic number theory.
Compatible system of ℓ-adic representations
In number theory, a compatible system of ℓ-adic representations is an abstraction of certain important families of ℓ-adic Galois representations, indexed by prime numbers ℓ, that have compatibility pr
Group cohomology
In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous
Algebraic number field
In mathematics, an algebraic number field (or simply number field) is an extension field of the field of rational numbers such that the field extension has finite degree (and hence is an algebraic fie
Eisenstein sum
In mathematics, an Eisenstein sum is a finite sum depending on a finite field and related to a Gauss sum. Eisenstein sums were introduced by Eisenstein in 1848, named "Eisenstein sums" by Stickelberge
Class formation
In mathematics, a class formation is a topological group acting on a module satisfying certain conditions. Class formations were introduced by Emil Artin and John Tate to organize the various Galois g
Hermite's problem
Hermite's problem is an open problem in mathematics posed by Charles Hermite in 1848. He asked for a way of expressing real numbers as sequences of natural numbers, such that the sequence is eventuall
Finite extensions of local fields
In algebraic number theory, through completion, the study of ramification of a prime ideal can often be reduced to the case of local fields where a more detailed analysis can be carried out with the a
Kummer–Vandiver conjecture
In mathematics, the Kummer–Vandiver conjecture, or Vandiver conjecture, states that a prime p does not divide the class number hK of the maximal real subfield of the p-th cyclotomic field. The conject
Biquadratic field
In mathematics, a biquadratic field is a number field K of a particular kind, which is a Galois extension of the rational number field Q with Galois group the Klein four-group.
Cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to Q, the field of rational numbers. Cyclotomic fields played a crucial role in the development of
Conductor of an abelian variety
In mathematics, in Diophantine geometry, the conductor of an abelian variety defined over a local or global field F is a measure of how "bad" the bad reduction at some prime is. It is connected to the
Monogenic field
In mathematics, a monogenic field is an algebraic number field K for which there exists an element a such that the ring of integers OK is the subring Z[a] of K generated by a. Then OK is a quotient of
Dedekind domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown
Supersingular prime (algebraic number theory)
In algebraic number theory, a supersingular prime for a given elliptic curve is a prime number with a certain relationship to that curve. If the curve E is defined over the rational numbers, then a pr
Galois cohomology
In 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. A Galois group G associated to a
Rigid analytic space
In mathematics, a rigid analytic space is an analogue of a complex analytic space over a nonarchimedean field. Such spaces were introduced by John Tate in 1962, as an outgrowth of his work on uniformi
Maillet's determinant
In mathematics, Maillet's determinant Dp is the determinant of the matrix introduced by whose entries are R(s/r) for s,r = 1, 2, ..., (p – 1)/2 ∈ Z/pZ for an odd prime p, where and R(a) is the least p
Herbrand quotient
In mathematics, the Herbrand quotient is a quotient of orders of cohomology groups of a cyclic group. It was invented by Jacques Herbrand. It has an important application in class field theory.
Abelian extension
In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other
Fundamental discriminant
In mathematics, a fundamental discriminant D is an integer invariant in the theory of integral binary quadratic forms. If Q(x, y) = ax2 + bxy + cy2 is a quadratic form with integer coefficients, then
Elliptic unit
In mathematics, elliptic units are certain units of abelian extensions of imaginary quadratic fields constructed using singular values of modular functions, or division values of elliptic functions. T
Reciprocity law
In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials with integer coefficients. Recall that first reciprocity law, quadr
Discriminant of an algebraic number field
In mathematics, the discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the (ring of integers of the) algebraic number field. More specifica
Quadratic field
In algebraic number theory, a quadratic field is an algebraic number field of degree two over Q, the rational numbers. Every such quadratic field is some Q(√d) where d is a (uniquely defined) square-f
Galois extension
In mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed by the automorphism group Aut(E/F) is precis
Fractional ideal
In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In so
Class number formula
In number theory, the class number formula relates many important invariants of a number field to a special value of its Dedekind zeta function.
Rational reciprocity law
In number theory, a rational reciprocity law is a reciprocity law involving residue symbols that are related by a factor of +1 or –1 rather than a general root of unity. As an example, there are ratio
Brumer–Stark conjecture
The Brumer–Stark conjecture is a conjecture in algebraic number theory giving a rough generalization of both the analytic class number formula for Dedekind zeta functions, and also of Stickelberger's
Basic Number Theory
Basic Number Theory is an influential book by André Weil, an exposition of algebraic number theory and class field theory with particular emphasis on valuation-theoretic methods. Based in part on a co
Artin's conjecture on primitive roots
In number theory, Artin's conjecture on primitive roots states that a given integer a that is neither a square number nor −1 is a primitive root modulo infinitely many primes p. The conjecture also as
Class number problem
In mathematics, the Gauss class number problem (for imaginary quadratic fields), as usually understood, is to provide for each n ≥ 1 a complete list of imaginary quadratic fields (for negative integer
Power residue symbol
In algebraic number theory the n-th power residue symbol (for an integer n > 2) is a generalization of the (quadratic) Legendre symbol to n-th powers. These symbols are used in the statement and proof
Splitting of prime ideals in Galois extensions
In mathematics, the interplay between the Galois group G of a Galois extension L of a number field K, and the way the prime ideals P of the ring of integers OK factorise as products of prime ideals of
Cyclotomic unit
In mathematics, a cyclotomic unit (or circular unit) is a unit of an algebraic number field which is the product of numbers of the form (ζan − 1) for ζn an nth root of unity and 0 < a < n.
Lazard's universal ring
In mathematics, Lazard's universal ring is a ring introduced by Michel Lazard in over which the universal commutative one-dimensional formal group law is defined. There is a universal commutative one-
Modulus (algebraic number theory)
In mathematics, in the field of algebraic number theory, a modulus (plural moduli) (or cycle, or extended ideal) is a formal product of places of a global field (i.e. an algebraic number field or a gl
Greenberg's conjectures
Greenberg's conjecture is either of two conjectures in algebraic number theory proposed by Ralph Greenberg. Both are still unsolved as of 2021.
Local Fields
Corps Locaux by Jean-Pierre Serre, originally published in 1962 and translated into English as Local Fields by Marvin Jay Greenberg in 1979, is a seminal graduate-level algebraic number theory text co
Narrow class group
In algebraic number theory, the narrow class group of a number field K is a refinement of the class group of K that takes into account some information about embeddings of K into the field of real num
Twisted polynomial ring
In mathematics, a twisted polynomial is a polynomial over a field of characteristic in the variable representing the Frobenius map . In contrast to normal polynomials, multiplication of these polynomi
Parshin chain
In number theory, a Parshin chain is a higher-dimensional analogue of a place of an algebraic number field. They were introduced by in order to define an analogue of the idele class group for 2-dimens
Unit (ring theory)
In algebra, a unit of a ring is an invertible element for the multiplication of the ring. That is, an element u of a ring R is a unit if there exists v in R such that where 1 is the multiplicative ide
Euler system
In mathematics, an Euler system is a collection of compatible elements of Galois cohomology groups indexed by fields. They were introduced by Kolyvagin in his work on Heegner points on modular ellipti
Tate duality
In mathematics, Tate duality or Poitou–Tate duality is a duality theorem for Galois cohomology groups of modules over the Galois group of an algebraic number field or local field, introduced by John T
Galois module
In mathematics, a Galois module is a G-module, with G being the Galois group of some extension of fields. The term Galois representation is frequently used when the G-module is a vector space over a f
Ramification group
In number theory, more specifically in local class field theory, the ramification groups are a filtration of the Galois group of a local field extension, which gives detailed information on the ramifi
Norm group
In number theory, a norm group is a group of the form where is a finite abelian extension of nonarchimedean local fields. One of the main theorems in local class field theory states that the norm grou
Tate's thesis
In number theory, Tate's thesis is the 1950 PhD thesis of John Tate completed under the supervision of Emil Artin at Princeton University. In it, Tate used a translation invariant integration on the l
Leopoldt's conjecture
In algebraic number theory, Leopoldt's conjecture, introduced by H.-W. Leopoldt , states that the p-adic regulator of a number field does not vanish. The p-adic regulator is an analogue of the usual r
Elliptic Gauss sum
In mathematics, an elliptic Gauss sum is an analog of a Gauss sum depending on an elliptic curve with complex multiplication. The quadratic residue symbol in a Gauss sum is replaced by a higher residu
CM-field
In mathematics, a CM-field is a particular type of number field, so named for a close connection to the theory of complex multiplication. Another name used is J-field. The abbreviation "CM" was introd
Field norm
In mathematics, the (field) norm is a particular mapping defined in field theory, which maps elements of a larger field into a subfield.
Zahlbericht
In mathematics, the Zahlbericht (number report) was a report on algebraic number theory by Hilbert .
Unique factorization domain
In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem o
Adele ring
In mathematics, the adele ring of a global field (also adelic ring, ring of adeles or ring of adèles) is a central object of class field theory, a branch of algebraic number theory. It is the restrict
S-unit
In mathematics, in the field of algebraic number theory, an S-unit generalises the idea of unit of the ring of integers of the field. Many of the results which hold for units are also valid for S-unit
Hilbert's twelfth problem
Kronecker's Jugendtraum or Hilbert's twelfth problem, of the 23 mathematical Hilbert problems, is the extension of the Kronecker–Weber theorem on abelian extensions of the rational numbers, to any bas
Ideal class group
In number theory, the ideal class group (or class group) of an algebraic number field K is the quotient group JK/PK where JK is the group of fractional ideals of the ring of integers of K, and PK is i