Basic topics

• Algebraic number field
  Algebraic number field
  In mathematics, an algebraic number field F is a finite field extension of the field of rational numbers Q...
  
  
  • Gaussian integer
    Gaussian integer
    In number theory, a Gaussian integer is a complex number whose real and imaginary part are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[i]. The Gaussian integers are a special case of the quadratic...
    
    , Gaussian rational
    Gaussian rational
    In mathematics, a Gaussian rational number is a complex number of the form p + qi, where p and q are both rational numbers....
    
    
  • Quadratic field
    Quadratic field
    In algebraic number theory, a quadratic field is an algebraic number field K of degree two over Q. It is easy to show that the map d ↦ Q is a bijection from the set of all square-free integers d ≠ 0, 1 to the set of all quadratic fields...
    
    
  • Cyclotomic field
    Cyclotomic field
    In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers...
    
    
  • Cubic field
    Cubic field
    In mathematics, specifically the area of algebraic number theory, a cubic field is an algebraic number field of degree three.-Definition:If K is a field extension of the rational numbers Q of degree [K:Q] = 3, then K is called a cubic field...
    
    
  • Biquadratic field
    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. Such fields are all obtained by adjoining two square roots. Therefore in explicit terms we havefor rational numbers a and b...
    
    
• Quadratic reciprocity
  Quadratic reciprocity
  In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic which gives conditions for the solvability of quadratic equations modulo prime numbers...
  
  
• Ideal class group
  Ideal class group
  In mathematics, the extent to which unique factorization fails in the ring of integers of an algebraic number field can be described by a certain group known as an ideal class group...
  
  
• Dirichlet's unit theorem
  Dirichlet's unit theorem
  In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring OK of algebraic integers of a number field K...
  
  
• Discriminant of an algebraic number field
  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 algebraic number field...
  
  
• Ramification
  Ramification
  In mathematics, ramification is a geometric term used for 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign...
  
  
• Root of unity
  Root of unity
  In mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete...
  
  
• Gaussian period
  Gaussian period
  In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of roots of unity. The periods permit explicit calculations in cyclotomic fields connected with Galois theory and with harmonic analysis . They are basic in the classical theory called cyclotomy...
  
  , Gauss sum
  Gauss sum
  In mathematics, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typicallyG := G= \sum \chi\cdot \psi...
  
  

Important problems

• Fermat's last theorem
  Fermat's Last Theorem
  In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
  
  
• Class number problem for imaginary quadratic fields
  Class number problem for imaginary quadratic fields
  In mathematics, the Gauss class number problem , as usually understood, is to provide for each n ≥ 1 a complete list of imaginary quadratic fields with class number n. It is named after the great mathematician Carl Friedrich Gauss. It can also be stated in terms of discriminants...
  
  
  • Stark–Heegner theorem
    Stark–Heegner theorem
    In number theory, a branch of mathematics, the Stark–Heegner theorem states precisely which quadratic imaginary number fields admit unique factorisation in their ring of integers...
    
    
    • Heegner number
      Heegner number
      In number theory, a Heegner number is a square-free positive integer d such that the imaginary quadratic field Q has class number 1...
      
      
• Langlands program
  Langlands program
  The Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. It was proposed by ....
  
  

General aspects

• Different ideal
  Different ideal
  In algebraic number theory, the different ideal is defined to account for the lack of duality in the ring of integers of an algebraic number field K, with respect to the field trace...
  
  
• Dedekind domain
  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 that such a factorization is then necessarily unique up to the order of the factors...
  
  
• Splitting of prime ideals in Galois extensions
  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 OL, provides one of the richest parts of algebraic number theory...
  
  
• Decomposition group
• Inertia group
• Frobenius automorphism
• Chebotarev's density theorem
  Chebotarev's density theorem
  Chebotarev's density theorem in algebraic number theory describes statistically the splitting of primes in a given Galois extension K of the field Q of rational numbers. Generally speaking, a prime integer will factor into several ideal primes in the ring of algebraic integers of K. There are only...
  
  
• Totally real field
• Local field
  Local field
  In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...
  
  
  • p-adic number
    P-adic number
    In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...
    
    
  • p-adic analysis
    P-adic analysis
    In mathematics, p-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of p-adic numbers....
    
    
• Adele ring
  Adele ring
  In algebraic number theory and topological algebra, the adele ring is a topological ring which is built on the field of rational numbers . It involves all the completions of the field....
  
  
• Idele group
• Idele class group
• Adelic algebraic group
  Adelic algebraic group
  In abstract algebra, an adelic algebraic group is a topological group defined by an algebraic group G over a number field K, and the adele ring A = A of K. It consists of the points of G having values in A; the definition of the appropriate topology is straightforward only in case G is a linear...
  
  
• Global field
  Global field
  In mathematics, the term global field refers to either of the following:*an algebraic number field, i.e., a finite extension of Q, or*a global function field, i.e., the function field of an algebraic curve over a finite field, equivalently, a finite extension of Fq, the field of rational functions...
  
  
• Hasse principle
  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 piece together solutions modulo powers of each different prime number...
  
  
  • Hasse–Minkowski theorem
    Hasse–Minkowski theorem
    The Hasse–Minkowski theorem is a fundamental result in number theory which states that two quadratic forms over a number field are equivalent if and only if they are equivalent locally at all places, i.e. equivalent over every completion of the field...
    
    
• Galois module
  Galois module
  In mathematics, a Galois module is a G-module where G is 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 field or a free module over a ring, but can also be used as a synonym for G-module...
  
  
• Galois cohomology
  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...
  
  
  • Brauer group
    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 of finite rank over K and addition is induced by the tensor product of algebras. It arose out of attempts to classify division algebras over a field and is...
    
    

Class field theory

• Class field theory
  Class field theory
  In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...
  
  
• Abelian extension
  Abelian extension
  In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is a cyclic group, we have a cyclic extension. More generally, a Galois extension is called solvable if its Galois group is solvable....
  
  
• Kronecker–Weber theorem
  Kronecker–Weber theorem
  In algebraic number theory, the Kronecker–Weber theorem states that every finite abelian extension of the field of rational numbers Q, or in other words, every algebraic number field whose Galois group over Q is abelian, is a subfield of a cyclotomic field, i.e. a field obtained by adjoining a root...
  
  
• Hilbert class field
  Hilbert class field
  In algebraic number theory, the Hilbert class field E of a number field K is the maximal abelian unramified extension of K. Its degree over K equals the class number of K and the Galois group of E over K is canonically isomorphic to the ideal class group of K using Frobenius elements for prime...
  
  
• Takagi existence theorem
  Takagi existence theorem
  In class field theory, the Takagi existence theorem states that for any number field K there is a one-to-one inclusion reversing correspondence between the finite abelian extensions of K and the generalized ideal class groups defined via a modulus of K.It is called an existence theorem because a...
  
  
• Hasse norm theorem
  Hasse norm theorem
  In number theory, the Hasse norm theorem states that if L/K is a cyclic extension of number fields, then if a nonzero element of K is a local norm everywhere, then it is a global norm....
  
  
• Artin reciprocity
  Artin reciprocity
  The Artin reciprocity law, established by Emil Artin in a series of papers , is a general theorem in number theory that forms a central part of the global class field theory...
  
  
• Local class field theory

Iwasawa theory

• Iwasawa theory
  Iwasawa theory
  In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa, in the 1950s, as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur...
  
  
• Herbrand–Ribet theorem
  Herbrand–Ribet theorem
  In mathematics, the Herbrand–Ribet theorem is a result on the class number of certain number fields. It is a strengthening of Ernst Kummer's theorem to the effect that the prime p divides the class number of the cyclotomic field of p-th roots of unity if and only if p divides the numerator of the...
  
  
• Vandiver's conjecture
• Stickelberger's theorem
  Stickelberger's theorem
  In mathematics, Stickelberger's theorem is a result of algebraic number theory, which gives some information about the Galois module structure of class groups of cyclotomic fields...
  
  
• Euler system
  Euler system
  In mathematics, an Euler system is collection of compatible elements of Galois cohomology groups indexed by fields. They were introduced by in his work on Heegner points on modular elliptic curves, which was motivated by his earlier paper and the work of...
  
  
• p-adic L-function
  P-adic L-function
  In mathematics, a p-adic zeta function, or more generally a p-adic L-function, is a function analogous to the Riemann zeta function, or more general L-functions, but whose domain and target are p-adic...
  
  

Arithmetic geometry

• Arithmetic geometry
• Complex multiplication
  Complex multiplication
  In mathematics, complex multiplication is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense In mathematics, complex multiplication is the...
  
  
• Abelian variety of CM-type
  Abelian variety of CM-type
  In mathematics, an abelian variety A defined over a field K is said to have CM-type if it has a large enough commutative subring in its endomorphism ring End. The terminology here is from complex multiplication theory, which was developed for elliptic curves in the nineteenth century...
  
  
  • Chowla–Selberg formula
• Hasse–Weil zeta function