Ring theory
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List of abstract algebra topics
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Outline of abstract algebra
Field theory
Field theory may refer to:
Algebraic number theory
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Class field theory
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Field extensions
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Finite field
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Galois theory
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Real closed field
Algebraic number theory
Algebraic number theory is a major branch of number theory that studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O...
Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields and function fields of curves over finite fields and arithmetic propert...
Field extensions
Finite field
In algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains a finite number of elements, called its order (the size of the underlying set). As with any ...
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. Using Galois theory, certain problems in...
Galois theory - Wikipedia
Real closed field
In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers...
Properties of polynomial roots
In mathematics, a polynomial is an expression of the formwhere the ai belong to some field, which, in this article, is always the field of the complex numbers. The natural number n is known as the de...
Separable polynomial
In mathematics, a polynomial P(X) over a given field K is separable if its roots are distinct in an algebraic closure of K, that is, the number of its distinct roots is equal to its degree.This concep...
Embedding problem
In Galois theory, a branch of mathematics, the embedding problem is a generalization of the inverse Galois problem. Roughly speaking, it asks whether a given Galois extension can be embedded into a Ga...
Stark conjectures
In number theory, the Stark conjectures, introduced by Stark (1971, 1975, 1976, 1980) and later expanded by Tate (1984), give conjectural information about the coefficient of the leading ...
Resolvent (Galois theory)
In Galois theory, a discipline within the field of abstract algebra, a resolvent for a permutation group G is a polynomial whose coefficients depend polynomially on the coefficients of a given polynom...
Perfect field
In algebra, a field k is said to be perfect if any one of the following equivalent conditions holds:Otherwise, k is called imperfect.In particular, all fields of characteristic zero and all finite fie...
Equally spaced polynomial
An equally spaced polynomial (ESP) is a polynomial used in finite fields, specifically GF(2) (binary). An s-ESP of degree sm can be written as:or
Over GF(2) the ESP has many interesting propertie...
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 Iwasawa...
Liouville's theorem (differential algebra)
In mathematics, Liouville's theorem, originally formulated by Joseph Liouville in the 1830s and 1840s, places an important restriction on antiderivatives that can be expressed as elementary functions....
Liouville's theorem (differential algebra) - Wikipedia
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....
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. The concept wa...
Rational variety
In mathematics, a rational variety is an algebraic variety, over a given field K, which is birationally equivalent to a projective space of some dimension over K. This means that its function field is...
15 and 290 theorems
The 15 theorem or Conway–Schneeberger Fifteen Theorem, proved by John H. Conway and W. A. Schneeberger in 1993, states that if a positive definite quadratic form with integer matrix represents all pos...
Kronecker–Weber theorem
In algebraic number theory, it can be shown that every cyclotomic field is an abelian extension of the rational number field Q. The Kronecker–Weber theorem provides a partial converse: every abelian e...
Tensor product of fields
In abstract algebra, the theory of fields lacks a direct product: the direct product of two fields, considered as a ring is never itself a field. On the other hand it is often required to 'join' two f...
Field (mathematics)
In abstract algebra, a field is a nonzero commutative ring that contains a multiplicative inverse for every nonzero element, or equivalently a ring whose nonzero elements form an abelian group under m...
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.
Let K be a field and L a finite extension (and hence an alg...
Half-integer
In mathematics, a half-integer is a number of the form where is an integer. For example, are all half-integers.Half-integers occur frequently enough in mathematical contexts that a special term for ...
Fundamental theorem of algebra
The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients,...
Ground field
In mathematics, a ground field is a field K fixed at the beginning of the discussion.
It is used in various areas of algebra:
Ground field - Wikipedia
Complete field
In mathematics, a complete field is a field equipped with a metric and complete with respect to that metric. Basic examples include the real numbers, the complex numbers, and complete valued fields (s...
Regular extension
In field theory, a branch of algebra, a field extension is said to be regular if k is algebraically closed in L (i.e., where is the set of elements in L algebraic over k) and L is separable over k...
Algebraically closed field
In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F[x], the ring of polynomials in the variable x with coefficients in F.
As an example, the...
Jacobson–Bourbaki theorem
In algebra, the Jacobson–Bourbaki theorem is a theorem used to extend Galois theory to field extensions that need not be separable. It was introduced by Nathan Jacobson (1944) for commutativ...