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Construction of a larger algebraic field by "adding elements" to a smaller field
In mathematics, particularly in algebra, a field extension is a pair of fields K ⊆ L {\displaystyle K\subseteq L} , such that the operations of K are
Field_extension
Dimension of the extension field viewed as a vector space over the base field
mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. The concept plays an important
Degree_of_a_field_extension
Type of algebraic field extension
In field theory, a branch of algebra, an algebraic field extension E / F {\displaystyle E/F} is called a separable extension if for every α ∈ E {\displaystyle
Separable_extension
Mathematical group
field extension is a symmetry group characterizing how it extends the base field. Each element of the Galois group is a transformation of the field extension
Galois_group
Algebraic field extension
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
Galois_extension
Extension of a mathematical field with polynomial roots
mathematics, an algebraic extension is a field extension L/K such that every element of the larger field L is algebraic over the smaller field K; that is, every
Algebraic_extension
Type of algebraic field extension
In abstract algebra, a normal extension is an algebraic field extension L/K for which every irreducible polynomial over K that has a root in L splits
Normal_extension
Field extension that is not algebraic
mathematics, a transcendental extension L / K {\displaystyle L/K} is a field extension such that there exists an element in the field L {\displaystyle L} that
Transcendental_extension
Algebraic structure with addition, multiplication, and division
symmetries of field extensions, provides an elegant proof of the Abel–Ruffini theorem that general quintic equations cannot be solved in radicals. Fields serve
Field_(mathematics)
Algebraic structure
irreducible polynomial over K {\displaystyle K} has no multiple roots in any field extension L / K {\displaystyle L/K} . Every irreducible polynomial over K {\displaystyle
Perfect_field
Mathematical field obtained by adjunction of nth roots
specifically in field theory, a radical extension of a field K {\displaystyle K} is a field extension obtained by a tower of field extensions, each generated
Radical_extension
Field theory is the branch of algebra that studies fields
then E is an extension field of F. We then also say that E/F is a field extension. Degree of an extension Given an extension E/F, the field E can be considered
Glossary_of_field_theory
Finite extension of the rationals
mathematics, an algebraic number field (or simply number field) is an extension field K {\displaystyle K} of the field of rational numbers Q {\displaystyle
Algebraic_number_field
Topics referred to by the same term
theory of extension, in geometry Field extension, in Galois theory Group extension, in abstract algebra and homological algebra Homotopy extension property
Extension
Function in algebra
be a field extension of K. An extension of v (to L) is a valuation w of L such that the restriction of w to K is v. The set of all such extensions is studied
Valuation_(algebra)
Galois extension whose Galois group is abelian
finite extension of a finite field is a cyclic extension. Class field theory provides detailed information about the abelian extensions of number fields, function
Abelian_extension
Alebraic concept
In algebra, a purely inseparable extension of fields is an extension k ⊆ K of fields of characteristic p > 0 such that every element of K is a root of
Purely_inseparable_extension
Field extension generated by a one element
In field theory, a simple extension is a field extension that is generated by the adjunction of a single element, called a primitive element. Simple extensions
Simple_extension
Mathematical function
the field trace is a particular function defined with respect to a finite field extension L/K, which is a K-linear map from L onto K. Let K be a field and
Field_trace
Concept in abstract algebra
In field theory, a branch of mathematics, the minimal polynomial of an element α {\displaystyle \alpha } of an extension field of a field is, roughly speaking
Minimal polynomial (field theory)
Minimal_polynomial_(field_theory)
Filtration of the Galois group of a local field extension
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
Ramification_group
Concept in field theory mathematics
a subfield. Let K be a field and L a finite extension (and hence an algebraic extension) of K. The field L is then a finite-dimensional vector space over
Field_norm
Theoretical object in mathematics
a curve over a field with one element. By 1991, Smirnov had taken some steps towards algebraic geometry over F1, introducing extensions of F1 and using
Field_with_one_element
Algebraic structure where all polynomials have roots
xn − 1. A field extension that is contained in an extension generated by the roots of unity is a cyclotomic extension, and the extension of a field generated
Algebraically_closed_field
Finitely generated extension field of positive transcendence degree
field (often abbreviated as function field) of n {\displaystyle n} variables over a field k {\displaystyle k} is a finitely generated field extension
Algebraic_function_field
Field generated by all rupture-fields of a polynomial over a field
abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial splits
Splitting_field
Algebraic field extension
particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures
Algebraic_closure
Study of differential field extensions induced by linear differential equations
differential field extension generated by the solutions of a linear differential equation, using the differential Galois group of the field extension. A major
Picard–Vessiot_theory
Branch of algebraic number theory concerned with abelian extensions
mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local
Class_field_theory
Given a finite field extension K / F {\displaystyle K/F} over a locally compact field F {\displaystyle F} , there is at most one unique field norm | ⋅ | K
Locally_compact_field
Polynomial without nontrivial factorization
appear naturally in the study of polynomial factorization and algebraic field extensions. It is helpful to compare irreducible polynomials to prime numbers:
Irreducible_polynomial
Mathematical concept
kinds of global fields: Algebraic number field: A finite extension of Q {\displaystyle \mathbb {Q} } Global function field: The function field of an irreducible
Global_field
Correspondence between subfields and subgroups
theory is a result that describes the structure of certain types of field extensions in relation to groups. It was proved by Évariste Galois in his development
Fundamental theorem of Galois theory
Fundamental_theorem_of_Galois_theory
In field theory, a branch of algebra, a primary extension L of K is a field extension such that the algebraic closure of K in L is purely inseparable over
Primary_extension
Mathematical connection between field theory and group theory
theory using number fields, finite fields or local fields as the base field. It allows one to more easily study infinite extensions. Again this is important
Galois_theory
Type of field extension
In field theory, a branch of algebra, a field extension L / k {\displaystyle L/k} is said to be regular if k is algebraically closed in L (i.e., k = k
Regular_extension
Ancient geometric construction problem
original pair of points (0,0) and (1,0). As every field extension has degree 2 or 1, and as the field extension over Q {\displaystyle \mathbb {Q} } of the coordinates
Doubling_the_cube
Concept in abstract algebra
algebraic extension. These notions generalize the algebraic numbers and the transcendental numbers (where the field extension is C/Q, with C being the field of
Algebraic_element
Theorem in field theory
In field theory, a branch of mathematics, the isomorphism extension theorem is an important theorem regarding the extension of a field isomorphism to
Isomorphism_extension_theorem
Field in which every sum of two squares is a square
Pythagorean field is a field in which every sum of two squares is a square: equivalently it has a Pythagoras number equal to 1. A Pythagorean extension of a
Pythagorean_field
Topics referred to by the same term
stochastic theory a rule governing the degree of a field extension of a field extension in field theory This disambiguation page lists articles associated
Tower_rule
Mathematical element
are fields, then the notions of "integral over" and of an "integral extension" are precisely "algebraic over" and "algebraic extensions" in field theory
Integral_element
Field theory theorem
In field theory, the primitive element theorem states that every finite separable field extension is simple, i.e. generated by a single element. This theorem
Primitive_element_theorem
Theory in abstract algebra
description of certain types of field extensions involving the adjunction of nth roots of elements of the base field. The theory was originally developed
Kummer_theory
Mathematical topic
finite residue field. Let L / K {\displaystyle L/K} be a finite Galois extension of nonarchimedean local fields with finite residue fields ℓ / k {\displaystyle
Finite extensions of local fields
Finite_extensions_of_local_fields
applied in the context of a finite field extension L/K, by using the field trace. This requires the property that the field trace TrL/K provides a non-degenerate
Dual basis in a field extension
Dual_basis_in_a_field_extension
Generalization of quaternions to other fields
extending scalars (equivalently, tensoring with a field extension), i.e. for a suitable field extension K of F, A ⊗ F K {\displaystyle A\otimes _{F}K} is
Quaternion_algebra
Farm efficiency through education
Agricultural extension is the application of scientific research and new knowledge to agricultural practices through farmer education. The field of 'extension' now
Agricultural_extension
Algebraic structure
x^{p^{n}}-x=0} . Any finite field extension of a finite field is separable and simple. That is, if E {\displaystyle E} is a finite field and F {\displaystyle
Finite_field
Field extension of the rational numbers by a primitive root of unity
th root of unity. Then the n {\displaystyle n} th cyclotomic field is the field extension Q ( ζ n ) {\displaystyle \mathbb {Q} (\zeta _{n})} of Q {\displaystyle
Cyclotomic_field
Roots of an algebraic element's minimal polynomial
mathematics, in particular field theory, the conjugate elements or algebraic conjugates of an algebraic element α, over a field extension L/K, are the roots of
Conjugate element (field theory)
Conjugate_element_(field_theory)
About polynomials in several variables
because given a field K {\displaystyle \mathbb {K} } of characteristic zero, zero never divides the degree of the field extension K ( X ) / K ( F )
Jacobian_conjecture
Polynomial coprime with its derivative
Separable polynomials are used to define separable extensions: A field extension K ⊂ L is a separable extension if and only if for every α in L which is algebraic
Separable_polynomial
Mathematical concept in polynomial theory
have a common root (possibly in a field extension), or, equivalently, a common factor (possibly in a field extension as well). In some older texts, the
Resultant
Field extension of algebraic number field characterized by prime ideals of inertial deg 1
In mathematics, in the field of algebraic number theory, a Bauerian extension is a field extension of an algebraic number field which is characterized
Bauerian_extension
generalization to associative algebras of the notion of a separable field extension. A homomorphism of (unital, but not necessarily commutative) rings
Separable_algebra
the field extension is separable iff Ω L / K = 0. {\displaystyle \Omega _{L/K}=0.} Notice that this includes every perfect field: finite fields and fields
Smooth_morphism
In field theory, Steinitz's theorem states that a finite extension of fields L / K {\displaystyle L/K} is simple if and only if there are only finitely
Steinitz's theorem (field theory)
Steinitz's_theorem_(field_theory)
Algebraic surface in mathematics
10 Eckardt points is a Hilbert modular surface. Given a quadratic field extension K = Q ( p ) {\displaystyle K=\mathbb {Q} ({\sqrt {p}})} for p = 4 k
Hilbert_modular_variety
Major unsolved problem in transcendental number theory
conjecture is a conjecture about the transcendence degree of certain field extensions of the rational numbers Q {\displaystyle \mathbb {Q} } , which would
Schanuel's_conjecture
conductor of a finite abelian extension of local or global fields provides a quantitative measure of the ramification in the extension. The definition of the
Conductor (class field theory)
Conductor_(class_field_theory)
Square matrix constructed from a monic polynomial
{\displaystyle \lambda } of p ( x ) {\displaystyle p(x)} produces a field extension K = F ( λ ) ≅ F [ x ] / ( p ( x ) ) {\displaystyle K=F(\lambda )\cong
Companion_matrix
Number constructible via compass and straightedge
constructible number. This field is a field extension of the rational numbers and in turn is contained in the field of algebraic numbers. It is the Euclidean
Constructible_number
Topics referred to by the same term
Trivial extension may refer to the following types of extensions: A trivial field extension A trivial group extension A trivial algebra extension This disambiguation
Trivial_extension
System of numbers with non-finite quantities
each Cauchy sequence in the field converges. Equivalently, it has no proper dense ordered field extension. As an ordered field, it has a natural valuation
Levi-Civita_field
Ring produced from two fields
tensor product of two fields expresses in a single structure the different way to embed the two fields in a common extension field. First, one defines the
Tensor_product_of_fields
Converting interlaced video into a non-interlaced or progressive form
resolution) whereby 50i or 60i is converted to 25p or 30p. Field extension deinterlacing which takes each field (with only half the lines) and extend it to the entire
Deinterlacing
polynomials with coefficients in a finite field, in the field of rationals or in a finitely generated field extension of one of them. All factorization algorithms
Factorization of polynomials over finite fields
Factorization_of_polynomials_over_finite_fields
Aspect of algebraic number theory
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
Splitting of prime ideals in Galois extensions
Splitting_of_prime_ideals_in_Galois_extensions
étale algebra over a field is a special type of algebra, one that is isomorphic to a finite product of finite separable field extensions. An étale algebra
Étale_algebra
Topics referred to by the same term
of a local field extension Ramification theory of valuations, studies the set of extensions of a valuation v of a field K to an extension L of K Ramification
Ramification
Type of complex number
{\displaystyle \alpha _{2}} ... α n {\displaystyle \alpha _{n}} , the field extension Q ( α 1 , α 2 , . . . α n ) {\displaystyle \mathbb {Q} (\alpha _{1}
Algebraic_number
Topics referred to by the same term
the highest exponent Degree of a field extension Degree of an algebraic number field, its degree as a field extension of the rational numbers Degree of
Degree
Field in mathematics similar to the real numbers
algebraic extension of F. F is a formally real field such that no proper algebraic extension of F is formally real. (In other words, the field is maximal
Real_closed_field
Algebraic concept
algebra, a rupture field of a polynomial P ( X ) {\displaystyle P(X)} over a given field K {\displaystyle K} is a field extension of K {\displaystyle
Rupture_field
of algebraic number theory, a cubic field is an algebraic number field of degree three. If K is a field extension of the rational numbers Q of degree
Cubic_field
local class field theory (LCFT), introduced by Helmut Hasse, is the study of abelian extensions of local fields; here, "local field" means a field which is
Local_class_field_theory
In mathematics, non-abelian class field theory is a catchphrase, meaning the extension of the results of class field theory, the relatively complete and
Non-abelian class field theory
Non-abelian_class_field_theory
Topics referred to by the same term
element, an element of a field extension which is a root of some polynomial over the base field Algebraic extension, a field extension such that every element
Algebraic
Method for computing the relation of two integers with their greatest common divisor
compute the multiplicative inverse in algebraic field extensions and, in particular in finite fields of non-prime order. It follows that both extended
Extended_Euclidean_algorithm
Theory of a class of elliptic curves
generate the Hilbert class field H of K: the field extension degree [H:K] = h is the class number of K and the H/K is a Galois extension with Galois group isomorphic
Complex_multiplication
Roots of multiple multivariate polynomials
over some field k. A solution of a polynomial system is a set of values for the xis which belong to some algebraically closed field extension K of k, and
System of polynomial equations
System_of_polynomial_equations
Creating a "larger" Lie algebra from a smaller one, in one of several ways
algebra extension e is an enlargement of a given Lie algebra g by another Lie algebra h. Extensions arise in several ways. There is the trivial extension obtained
Lie_algebra_extension
Field (mathematics) generated by the square root of an integer
field extension can be accomplished using Minkowski's bound and the Kronecker symbol because of the finiteness of the class group. A quadratic field K
Quadratic_field
Group for which a given group is a normal subgroup
In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q {\displaystyle
Group_extension
Algebraic structure in linear algebra
with other structures. This is the case of algebras, which include field extensions, polynomial rings, associative algebras and Lie algebras. This is also
Vector_space
When an idea extends to a principal ideal in an extension of algebraic number fields
number field, which is not principal in that field, becomes principal after extension to the ring of integers of a larger algebraic number field. The study
Principalization_(algebra)
Theorem in algebraic number theory
that for any integer N there are only finitely many number fields, i.e., finite field extensions K of the rational numbers Q, such that the discriminant
Hermite–Minkowski_theorem
Branching out of a mathematical structure
set of extensions of a valuation of a field K to an extension field of K. This generalizes the notions in algebraic number theory, local fields, and Dedekind
Ramification_(mathematics)
(regular field extension of E) of smaller cardinality than F can be embedded in F. They were introduced by Ax (1968). Every hyper-finite field is a pseudo-finite
Hyper-finite_field
over a field which is a primary extension of K. If a field is weakly Ck, then any extension of transcendence degree n is weakly Ck+n. Any extension of an
Quasi-algebraically closed field
Quasi-algebraically_closed_field
Filename suffix indicating file type
class name, corresponding to today's name and extension. Internally, they were stored in different fields in the file system, but to ease entry, the user
Filename_extension
Operation in algebra
complexification, which is extension of scalars from the real numbers to the complex numbers. More generally, given any field extension K < L, one can extend
Change_of_rings
In mathematics, specifically class field theory, a ray class field is an abelian extension of a global field associated with a ray class group of ideal
Ray_class_field
Differential form in commutative algebra
be used to study the ramification in an extension of algebraic number fields. If L / K is a finite extension with rings of integers R and S respectively
Kähler_differential
for a field extension of characteristic p, introduced by Teichmüller (1936). Suppose k is a field of characteristic p and K is a field extension. A p-basis
P-basis
Measure of the size of the ring of integers
distinguish it from the relative discriminant of an extension K / L {\displaystyle K/L} of number fields. The latter is an ideal in the ring of integers of
Discriminant of an algebraic number field
Discriminant_of_an_algebraic_number_field
For finite field extension this can be explicitly found in Milne and for infinite extensions this follows since infinite Galois extensions are precisely
Composite_field_(mathematics)
Extension school of Harvard University
Harvard Extension School (HES) is the continuing education school of Harvard University, a private Ivy League research university in Cambridge, Massachusetts
Harvard_Extension_School
algebraic closure of K (necessarily separable because K is perfect). The field extension Ks/K is infinite, and the Galois group is accordingly given the Krull
Quasi-finite_field
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