Field extension, the Glossary

Table of Contents

1. 90 relations: Abelian extension, Abelian group, Addison-Wesley, Addition, Algebra, Algebraic closure, Algebraic fraction, Algebraic geometry, Algebraic independence, Algebraic number field, Algebraic number theory, Algebraic variety, Allyn & Bacon, Associative algebra, Automorphism, Azumaya algebra, Bijection, Brauer group, Cardinality of the continuum, Category of rings, Center (ring theory), Central simple algebra, Change of rings, Characteristic (algebra), Closure (mathematics), Complex number, Complexification, Constant function, Degree of a field extension, Dimension (vector space), Equivalence class, Field (mathematics), Field of fractions, Finite field, Function field of an algebraic variety, Fundamental theorem of Galois theory, Galois extension, Galois group, Galois theory, Generator (mathematics), Glossary of field theory, Group representation, Group ring, Ideal (ring theory), If and only if, Injective function, Irreducible polynomial, Isomorphism, Local ring, Mathematics, ... Expand index (40 more) »

2. Field extensions

Abelian extension

In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. Field extension and abelian extension are field extensions.

See Field extension and Abelian extension

Abelian group

In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.

See Field extension and Abelian group

Addison-Wesley

Addison–Wesley is an American publisher of textbooks and computer literature.

See Field extension and Addison-Wesley

Addition

Addition (usually signified by the plus symbol) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division.

See Field extension and Addition

Algebra

Algebra is the branch of mathematics that studies algebraic structures and the manipulation of statements within those structures.

See Field extension and Algebra

Algebraic closure

In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. Field extension and algebraic closure are field extensions.

See Field extension and Algebraic closure

Algebraic fraction

In algebra, an algebraic fraction is a fraction whose numerator and denominator are algebraic expressions.

See Field extension and Algebraic fraction

Algebraic geometry

Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems.

See Field extension and Algebraic geometry

Algebraic independence

In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation with coefficients in K. In particular, a one element set \ is algebraically independent over K if and only if \alpha is transcendental over K.

See Field extension and Algebraic independence

Algebraic number field

In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).

See Field extension and Algebraic number field

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.

See Field extension and Algebraic number theory

Algebraic variety

Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics.

See Field extension and Algebraic variety

Allyn & Bacon

Allyn & Bacon, founded in 1868, is a higher education textbook publisher in the areas of education, humanities and social sciences.

See Field extension and Allyn & Bacon

Associative algebra

In mathematics, an associative algebra A over a commutative ring (often a field) K is a ring A together with a ring homomorphism from K into the center of A. This is thus an algebraic structure with an addition, a multiplication, and a scalar multiplication (the multiplication by the image of the ring homomorphism of an element of K).

See Field extension and Associative algebra

Automorphism

In mathematics, an automorphism is an isomorphism from a mathematical object to itself.

See Field extension and Automorphism

Azumaya algebra

In mathematics, an Azumaya algebra is a generalization of central simple algebras to R-algebras where R need not be a field.

See Field extension and Azumaya algebra

Bijection

A bijection, bijective function, or one-to-one correspondence between two mathematical sets is a function such that each element of the first set (the domain) is mapped to exactly one element of the second set (the codomain).

See Field extension and Bijection

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.

See Field extension and Brauer group

Cardinality of the continuum

In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum.

See Field extension and Cardinality of the continuum

Category of rings

In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity).

See Field extension and Category of rings

Center (ring theory)

In algebra, the center of a ring R is the subring consisting of the elements x such that for all elements y in R. It is a commutative ring and is denoted as Z(R); 'Z' stands for the German word Zentrum, meaning "center".

See Field extension and Center (ring theory)

Central simple algebra

In ring theory and related areas of mathematics a central simple algebra (CSA) over a field K is a finite-dimensional associative ''K''-algebra A which is simple, and for which the center is exactly K. (Note that not every simple algebra is a central simple algebra over its center: for instance, if K is a field of characteristic 0, then the Weyl algebra K is a simple algebra with center K, but is not a central simple algebra over K as it has infinite dimension as a K-module.) For example, the complex numbers C form a CSA over themselves, but not over the real numbers R (the center of C is all of C, not just R).

See Field extension and Central simple algebra

Change of rings

In algebra, a change of rings is an operation of changing a coefficient ring to another.

See Field extension and Change of rings

Characteristic (algebra)

In mathematics, the characteristic of a ring, often denoted, is defined to be the smallest positive number of copies of the ring's multiplicative identity that will sum to the additive identity.

See Field extension and Characteristic (algebra)

Closure (mathematics)

In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset.

See Field extension and Closure (mathematics)

Complex number

In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted, called the imaginary unit and satisfying the equation i^.

See Field extension and Complex number

Complexification

In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include their scaling ("multiplication") by complex numbers.

See Field extension and Complexification

Constant function

In mathematics, a constant function is a function whose (output) value is the same for every input value.

See Field extension and Constant function

Degree of a field extension

In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. Field extension and degree of a field extension are field extensions.

See Field extension and Degree of a field extension

Dimension (vector space)

In mathematics, the dimension of a vector space V is the cardinality (i.e., the number of vectors) of a basis of V over its base field.

See Field extension and Dimension (vector space)

Equivalence class

In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes.

See Field extension and Equivalence class

Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers.

See Field extension and Field (mathematics)

Field of fractions

In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded.

See Field extension and Field of fractions

Finite field

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.

See Field extension and Finite field

Function field of an algebraic variety

In algebraic geometry, the function field of an algebraic variety V consists of objects that are interpreted as rational functions on V. In classical algebraic geometry they are ratios of polynomials; in complex geometry these are meromorphic functions and their higher-dimensional analogues; in modern algebraic geometry they are elements of some quotient ring's field of fractions.

See Field extension and Function field of an algebraic variety

Fundamental theorem of Galois theory

In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions in relation to groups.

See Field extension and Fundamental theorem of Galois theory

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 precisely the base field F. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory. Field extension and Galois extension are field extensions.

See Field extension and Galois extension

Galois group

In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension.

See Field extension and Galois group

Galois theory

In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory.

See Field extension and Galois theory

Generator (mathematics)

In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts.

See Field extension and Generator (mathematics)

Glossary of field theory

Field theory is the branch of mathematics in which fields are studied.

See Field extension and Glossary of field theory

Group representation

In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication.

See Field extension and Group representation

Group ring

In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group.

See Field extension and Group ring

Ideal (ring theory)

In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements.

See Field extension and Ideal (ring theory)

If and only if

In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements.

See Field extension and If and only if

Injective function

In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies.

See Field extension and Injective function

Irreducible polynomial

In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials.

See Field extension and Irreducible polynomial

Isomorphism

In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping.

See Field extension and Isomorphism

Local ring

In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on algebraic varieties or manifolds, or of algebraic number fields examined at a particular place, or prime.

See Field extension and Local ring

Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

See Field extension and Mathematics

Maximal ideal

In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all proper ideals.

See Field extension and Maximal ideal

Meromorphic function

In the mathematical field of complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all of D except for a set of isolated points, which are poles of the function.

See Field extension and Meromorphic function

Minimal polynomial (field theory)

In field theory, a branch of mathematics, the minimal polynomial of an element of an extension field of a field is, roughly speaking, the polynomial of lowest degree having coefficients in the smaller field, such that is a root of the polynomial.

See Field extension and Minimal polynomial (field theory)

Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus.

See Field extension and Modular arithmetic

Monic polynomial

In algebra, a monic polynomial is a non-zero univariate polynomial (that is, a polynomial in a single variable) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1.

See Field extension and Monic polynomial

Morphism

In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces.

See Field extension and Morphism

Multiplication

Multiplication (often denoted by the cross symbol, by the mid-line dot operator, by juxtaposition, or, on computers, by an asterisk) is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division.

See Field extension and Multiplication

Multiplicative inverse

In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1.

See Field extension and Multiplicative inverse

Normal 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 into linear factors in L. This is one of the conditions for an algebraic extension to be a Galois extension. Field extension and normal extension are field extensions.

See Field extension and Normal extension

Number theory

Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions.

See Field extension and Number theory

P-adic number

In number theory, given a prime number, the -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number rather than ten, and extending to the left rather than to the right.

See Field extension and P-adic number

Polynomial

In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms.

See Field extension and Polynomial

Polynomial ring

In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field.

See Field extension and Polynomial ring

Primary extension

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 K.Fried & Jarden (2008) p.44. Field extension and primary extension are field extensions.

See Field extension and Primary extension

Prime number

A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers.

See Field extension and Prime number

Primitive element theorem

In field theory, the primitive element theorem states that every finite separable field extension is simple, i.e. generated by a single element.

See Field extension and Primitive element theorem

Quotient group

A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored out").

See Field extension and Quotient group

Quotient ring

In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra.

See Field extension and Quotient ring

Rational function

In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials.

See Field extension and Rational function

Rational number

In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.

See Field extension and Rational number

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 isomorphic to the field of all rational functions for some set \ of indeterminates, where d is the dimension of the variety.

See Field extension and Rational variety

Real number

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature.

See Field extension and Real number

Regular extension

In field theory, a branch of algebra, a field extension L/k is said to be regular if k is algebraically closed in L (i.e., k. Field extension and regular extension are field extensions.

See Field extension and Regular extension

Restriction (mathematics)

In mathematics, the restriction of a function f is a new function, denoted f\vert_A or f, obtained by choosing a smaller domain A for the original function f. The function f is then said to extend f\vert_A.

See Field extension and Restriction (mathematics)

Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold.

See Field extension and Riemann surface

Ring (mathematics)

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist.

See Field extension and Ring (mathematics)

Ring homomorphism

In mathematics, a ring homomorphism is a structure-preserving function between two rings.

See Field extension and Ring homomorphism

Separable extension

In field theory, a branch of algebra, an algebraic field extension E/F is called a separable extension if for every \alpha\in E, the minimal polynomial of \alpha over is a separable polynomial (i.e., its formal derivative is not the zero polynomial, or equivalently it has no repeated roots in any extension field). Field extension and separable extension are field extensions.

See Field extension and Separable extension

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 distinct roots is equal to the degree of the polynomial.

See Field extension and Separable polynomial

Simple extension

In field theory, a simple extension is a field extension that is generated by the adjunction of a single element, called a primitive element. Field extension and simple extension are field extensions.

See Field extension and Simple extension

Simple ring

In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself.

See Field extension and Simple ring

Splitting field

In 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, i.e., decomposes into linear factors.

See Field extension and Splitting field

Subgroup

In group theory, a branch of mathematics, given a group under a binary operation ∗, a subset of is called a subgroup of if also forms a group under the operation ∗.

See Field extension and Subgroup

Subring

In mathematics, a subring of R is a subset of a ring that is itself a ring when binary operations of addition and multiplication on R are restricted to the subset, and that shares the same multiplicative identity as R. (Note that a subset of a ring R need not be a ring.) For those who define rings without requiring the existence of a multiplicative identity, a subring of R is just a subset of R that is a ring for the operations of R (this does imply it contains the additive identity of R).

See Field extension and Subring

Subset

In mathematics, a set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment).

See Field extension and Subset

Tower of fields

In mathematics, a tower of fields is a sequence of field extensions The name comes from such sequences often being written in the form A tower of fields may be finite or infinite. Field extension and tower of fields are field extensions.

See Field extension and Tower of fields

Transcendental extension

In mathematics, a transcendental extension L/K is a field extension such that there exists an element in the field L that is transcendental over the field K; that is, an element that is not a root of any univariate polynomial with coefficients in K. In other words, a transcendental extension is a field extension that is not algebraic.

See Field extension and Transcendental extension

Up to

Two mathematical objects and are called "equal up to an equivalence relation ".

See Field extension and Up to

Vector space

In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', can be added together and multiplied ("scaled") by numbers called ''scalars''.

See Field extension and Vector space

Zero of a function

In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) vanishes at x; that is, the function f attains the value of 0 at x, or equivalently, x is a solution to the equation f(x).

See Field extension and Zero of a function

See also

Field extensions

• Abelian extension
• Algebraic closure
• Algebraic extension
• Degree of a field extension
• Dual basis in a field extension
• Field extension
• Galois extension
• Normal extension
• Primary extension
• Regular extension
• Separable extension
• Simple extension
• Tower of fields

References

[1] https://en.wikipedia.org/wiki/Field_extension

Also known as Adjoining (field theory), Adjunction (field theory), Cubic extension, Cubic field extension, Degree (field theory), Extension field, Extension of a field, Finitely generated extension, Finitely generated field extension, Intermediate field, Purely transcendental, Purely transcendental extension, Quadratic extension, Quadratic field extension, Subextension, Subextension (field theory), Subfield (mathematics), Transcendental field extension, Trivial field extension.

, Maximal ideal, Meromorphic function, Minimal polynomial (field theory), Modular arithmetic, Monic polynomial, Morphism, Multiplication, Multiplicative inverse, Normal extension, Number theory, P-adic number, Polynomial, Polynomial ring, Primary extension, Prime number, Primitive element theorem, Quotient group, Quotient ring, Rational function, Rational number, Rational variety, Real number, Regular extension, Restriction (mathematics), Riemann surface, Ring (mathematics), Ring homomorphism, Separable extension, Separable polynomial, Simple extension, Simple ring, Splitting field, Subgroup, Subring, Subset, Tower of fields, Transcendental extension, Up to, Vector space, Zero of a function.