In mathematics, the discriminant of an algebraic number field is a numerical invariant containing information about ramified primes. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Interactive Mathematics Miscellany and Puzzles — A collection of articles on various math topics, with interactive Java... In mathematics, an algebraic number field (or simply number field) is a finite (and therefore algebraic) field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields... 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. ...

If K is an algebraic number field and R its ring of integers, the discriminant of K is associated to R and in some sense measures how large R is. In the special case of R = Z[α] for some algebraic integer α in K, it is simple to define, as the discriminant of the minimal polynomial P_α of α. This suffices, for example, in the case of the Gaussian integers: we take P(T) = T² + 1 for the choice α = i and calculate the discriminant as −4. In mathematics, an algebraic number relative to a field F is any element x of a given field K containing F such that x is a solution of a polynomial equation of the form a0xn + a1xn−1 + ··· + an −1x + an = 0 where n is a positive integer called the degree... In mathematics, an algebraic integer is a complex number α that is a root of an equation P(x) = 0 where P(x) is a monic polynomial (that is, the coefficient of the largest power of x in P(x) is one) with integer coefficients. ... The minimal polynomial of an n-by-n matrix A over a field F is the polynomial p(x) with leading coefficient 1 over F of least degree such that p(A)=0. ... A Gaussian integer is a complex number whose real and imaginary part are both integers. ...

This in fact works for any quadratic field or cyclotomic field; but certainly not in general. There we can only be sure that Z[α] can be chosen to be of finite index in R as an abelian group. This gives a factor (of the index) which is awkward to apply. The correct definition comes through a recognition that the discriminant of a polynomial is a square of a Vandermonde determinant, and that determinant is what we should generalise. The analogue in the general case is this: let the ω_i be an integral basis (i.e. basis for R as Z-module) and form In mathematics, a quadratic field is a field extension K/Q of the form where d is a non-zero rational number. ... In mathematics, the n-th roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... In linear algebra, a Vandermonde matrix is a matrix with a geometric progression in each row, i. ...

det(ω_i^(j))

where the superscripts mean that we take the conjugates. This (squared) leads to the correct general definition. In mathematics, in particular field theory, the conjugate elements of an algebraic element α, over a field K, are the (other) roots of the minimal polynomial PK,α(t) of α over K. If K is given inside an algebraically closed field C, then the conjugates can be taken inside...

Why this is the correct approach is best studied in terms of the real vector space A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...

and the embedding into it of R as a lattice. The determinant involved in the discriminant then has a simple interpretation as a volume of a fundamental region for R. See lattice for other meanings of this term, both within and without mathematics. ... Volume, also called capacity, is a quantification of how much space an object occupies. ... In mathematics, given a lattice Γ in a Lie group G, a fundamental domain is a set D of representatives for the cosets G/Γ, that is also a well-behaved set topologically, in a sense that can be made precise in one of several ways. ...

There is also a formula for the discriminant related to the quadratic form definition above, starting from the field trace. In the theory of Pontryagin duality for completions of K as local fields, the related different ideal occurs naturally in matching up Haar measures. This accounts for the role of the discriminant in the functional equation for the Dedekind zeta function, and thence in the analytic class number formula, and Brauer-Siegel theorem. In mathematics, the field trace is a linear mapping defined for certain field extensions. ... In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. ... In mathematics, a local field is a special type of field which has the additional property that it is a complete metric space with respect to a discrete valuation. ... In mathematics, the different ideal is defined in algebraic number theory, to account for the (possible) lack of duality in the ring of integers of an algebraic number field K, with respect to the field trace. ... In mathematical analysis, the Haar measure is a way to assign an invariant volume to subsets of locally compact topological groups and subsequently define an integral for functions on those groups. ... In mathematics or its applications, a functional equation is an equation in terms of independent variables, and also unknown functions, which are to be solved for. ... In mathematics, the Dedekind zeta function is a Dirichlet series defined for any algebraic number field K, and denoted ζK(s) where s is a complex variable. ... In number theory, the class number formula relates many important invariants of a number field to a special value of its Dedekind zeta function Let be a number field with , where denotes the number of real embeddings of , and is the number of complex embeddings of Let be the Dedekind...

A theorem of Stickelberger states that the discriminant D of an algebraic number field must be congruent to 0 or 1 modulo 4. A result of Kronecker is that D = 1 is possible only for the rational number field Q; this entails that every other number field has some ramified prime p in it. Lower bounds for discriminants, in terms of the degree, are proved by methods from the geometry of numbers and analytic number theory. In the case of an abelian extension, one of the results of class field theory (conductor-discriminant formula) is a factorisation of the discriminant according to characters, which in the case of an abelian extension of Q are Dirichlet characters. Modular arithmetic is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ... Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the natural numbers; all else is the work of man (Bell 1986, p. ... In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... 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. ... In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ... In number theory, the geometry of numbers is a topic and method arising from the work of Hermann Minkowski, on the relationship between convex sets and lattices in n-dimensional space. ... Analytic number theory is the branch of number theory that uses methods from mathematical analysis. ... In abstract algebra, an abelian extension is a field extension for which the associated Galois group is abelian. ... In mathematics, class field theory is a major branch of algebraic number theory. ... This article is about the mathematical concept. ... In number theory, a Dirichlet character is a function χ from the positive integers to the complex numbers which has the following properties: There exists a positive integer k such that χ(n) = χ(n + k) for all n. ...