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Field extension

Field extension

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In abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

, field extensions are the main object of study in field theory
Field theory (mathematics)
Field theory is a branch of mathematics which studies the properties of fields. A field is a mathematical entity for which addition, subtraction, multiplication and division are well-defined....

. The general idea is to start with a base field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

 and construct in some manner a larger field which contains the base field and satisfies additional properties. For instance, the set Q(√2) = {a + b√2 | a, b ∈ Q} is the smallest extension of Q which includes all solutions to the equation x2 = 2.

Definitions


Let L be a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

. If K is a subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

 of the underlying set of L which is closed
Closure (mathematics)
In mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a unique member of the same set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but...

 with respect to the field operations and inverses in L, then K is said to be a subfield of L, and L is said to be an extension field of K. We then say that L /K, read as "L over K", is a field extension.

If L is an extension of F which is in turn an extension of K, then F is said to be an intermediate field (or intermediate extension or subextension) of the field extension L /K.

Given a field extension L /K and a subset S of L, K(S) denotes the smallest subfield of L which contains K and S, a field generated by the adjunction
Adjunction (field theory)
In abstract algebra, adjunction is a construction in field theory, where for a given field extension E/F, subextensions between E and F are constructed.- Definition :...

 of elements of S to K. If S consists of only one element s, K(s) is a shorthand for K({s}). A field extension of the form L = K(s) is called a simple extension
Simple extension
In mathematics, more specifically in field theory, a simple extension is a field extension which is generated by the adjunction of a single element...

 and s is called a primitive element of the extension.

Given a field extension L /K, then L can also be considered as a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

 over K. The elements of L are the "vectors" and the elements of K are the "scalars", with vector addition and scalar multiplication obtained from the corresponding field operations. The dimension
Dimension (vector space)
In mathematics, the dimension of a vector space V is the cardinality of a basis of V. It is sometimes called Hamel dimension or algebraic dimension to distinguish it from other types of dimension...

 of this vector space is called the degree of the extension
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 extension. The concept plays an important role in many parts of mathematics, including algebra and number theory — indeed in any area where fields appear prominently.-...

, and is denoted by [L : K].

An extension of degree 1 (that is, one where L is equal to K) is called a trivial extension. Extensions of degree 2 and 3 are called quadratic extensions and cubic extensions, respectively. Depending on whether the degree is finite or infinite the extension is called a finite extension or infinite extension.

Examples


The field of complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s C is an extension field of the field of real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s R, and R in turn is an extension field of the field of rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

s Q. Clearly then, C/Q is also a field extension. We have [C : R] = 2 because {1,i} is a basis, so the extension C/R is finite. This is a simple extension because C=R(). [R : Q] = (the cardinality of the continuum
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. It is an infinite cardinal number and is denoted by |\mathbb R| or \mathfrak c ....

), so this extension is infinite.

The set Q(√2) = {a + b√2 | a, b ∈ Q} is an extension field of Q, also clearly a simple extension. The degree is 2 because {1, √2} can serve as a basis. Finite extensions of Q are also called 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...

s and are important in number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

.

Another extension field of the rationals, quite different in flavor, is the field of 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...

s Qp for a prime number p.

It is common to construct an extension field of a given field K as a quotient ring
Quotient ring
In ring theory, a branch of modern algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linear algebra...

 of the polynomial ring
Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...

 K[X] in order to "create" a root for a given polynomial f(X). Suppose for instance that K does not contain any element x with x2 = −1. Then the polynomial X2 + 1 is irreducible
Irreducible polynomial
In mathematics, the adjective irreducible means that an object cannot be expressed as the product of two or more non-trivial factors in a given set. See also factorization....

 in K[X], consequently the ideal
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....

 (X2 + 1) generated by this polynomial is maximal
Maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal which is maximal amongst all proper ideals. In other words, I is a maximal ideal of a ring R if I is an ideal of R, I ≠ R, and whenever J is another ideal containing I as a subset, then either J = I or J = R...

, and L = K[X]/(X2 + 1) is an extension field of K which does contain an element whose square is −1 (namely the residue class of X).

By iterating the above construction, one can construct a splitting field
Splitting field
In abstract algebra, a splitting field of a polynomial with coefficients in a field is a smallest field extension of that field over which the polynomial factors into linear factors.-Definition:...

 of any polynomial from K[X]. This is an extension field L of K in which the given polynomial splits into a product of linear factors.

If p is any prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

 and n is a positive integer, we have a finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...

 GF(pn) with pn elements; this is an extension field of the finite field GF(p) = Z/pZ with p elements.

Given a field K, we can consider the field K(X) of all rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...

s in the variable X with coefficients in K; the elements of K(X) are fractions of two polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

s over K, and indeed K(X) is the field of fractions
Field of fractions
In abstract algebra, the field of fractions or field of quotients of an integral domain is the smallest field in which it can be embedded. The elements of the field of fractions of the integral domain R have the form a/b with a and b in R and b ≠ 0...

 of the polynomial ring K[X]. This field of rational functions is an extension field of K. This extension is infinite.

Given a Riemann surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...

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

s defined on M is a field, denoted by C(M). It is an extension field of C, if we identify every complex number with the corresponding constant function
Constant function
In mathematics, a constant function is a function whose values do not vary and thus are constant. For example the function f = 4 is constant since f maps any value to 4...

 defined on M.

Given an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...

 V over some field K, then the function field
Function field of an algebraic variety
In algebraic geometry, the function field of an algebraic variety V consists of objects which are interpreted as rational functions on V...

 of V, consisting of the rational functions defined on V and denoted by K(V), is an extension field of K.

Elementary properties


If L/K is a field extension, then L and K share the same 0 and the same 1. The additive group (K,+) is a subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...

 of (L,+), and the multiplicative group (K−{0},·) is a subgroup of (L−{0},·). In particular, if x is an element of K, then its additive inverse −x computed in K is the same as the additive inverse of x computed in L; the same is true for multiplicative inverses of non-zero elements of K.

In particular then, the characteristics
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...

 of L and K are the same.

Algebraic and transcendental elements


If L is an extension of K, then an element of L which is a root of a nonzero polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

 over K is said to be algebraic
Algebraic element
In mathematics, if L is a field extension of K, then an element a of L is called an algebraic element over K, or just algebraic over K, if there exists some non-zero polynomial g with coefficients in K such that g=0...

 over K. Elements that are not algebraic are called transcendental. As an example:
  • In C/R, i is algebraic because it is a root of x2 + 1.
  • In R/Q, √2 + √3 is algebraic, because it is a root of x4 − 10x2 + 1
  • In R/Q, e is transcendental because there is no polynomial with rational coefficients that has e as a root (see transcendental number
    Transcendental number
    In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...

    )
  • In C/R, e is algebraic because it is the root of x − e

The special case of C/Q is especially important, and the names algebraic number
Algebraic number
In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...

 and transcendental number are used to describe the complex numbers that are algebraic and transcendental (respectively) over Q.

If every element of L is algebraic over K, then the extension L/K is said to be an algebraic extension
Algebraic extension
In abstract algebra, a field extension L/K is called algebraic if every element of L is algebraic over K, i.e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i.e...

; otherwise it is said to be transcendental.
A subset S of L is called algebraically independent over K if no non-trivial polynomial relation with coefficients in K exists among the elements of S. The largest cardinality of an algebraically independent set is called the transcendence degree
Transcendence degree
In abstract algebra, the transcendence degree of a field extension L /K is a certain rather coarse measure of the "size" of the extension...

 of L/K. It is always possible to find a set S, algebraically independent over K, such that L/K(S) is algebraic. Such a set S is called a transcendence basis of L/K. All transcendence bases have the same cardinality, equal to the transcendence degree of the extension. An extension L/K is said to be purely transcendental if and only if there exists a transcendence basis S of L/K such that L=K(S). Such an extension has the property that all elements of L except those of K are transcendental over K, but, however, there are extensions with this property which are not purely transcendental. In addition, if L/K is purely transcendental and S is a transcendence basis of the extension, it doesn't necessarily follow that L=K(S). (For example, consider the extension Q(x,√x)/Q, where x is transcendental over Q. The set {x} is algebraically independent since x is transcendental. Obviously, the extension Q(x,√x)/Q(x) is algebraic, hence {x} is a transcendence basis. It doesn't generate the whole extension because there is no polynomial expression in x for √x. But it is easy to see that {√x} is a transcendence basis that generates Q(x,√x)), so this extension is indeed purely transcendental.)

It can be shown that an extension is algebraic if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

 it is the
union of its finite subextensions. In particular, every finite extension is algebraic. For example,
  • C/R and Q(√2)/Q, being finite, are algebraic.
  • R/Q is transcendental, although not purely transcendental.
  • K(X)/K is purely transcendental.


A simple extension is finite if generated by an algebraic element, and purely transcendental if generated by a transcendental element. So
  • R/Q is not simple, as it is neither finite nor purely transcendental.


Every field K has an algebraic closure
Algebraic closure
In mathematics, 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 in mathematics....

; this is essentially the largest extension field of K which is algebraic over K and which contains all roots of all polynomial equations with coefficients in K. For example, C is the algebraic closure of R.

Normal, separable and Galois extensions


An algebraic extension L/K is called normal
Normal extension
In abstract algebra, an algebraic field extension L/K is said to be normal if L is the splitting field of a family of polynomials in K[X]...

 if every irreducible polynomial
Irreducible polynomial
In mathematics, the adjective irreducible means that an object cannot be expressed as the product of two or more non-trivial factors in a given set. See also factorization....

 in K[X] that has a root in L completely factors into linear factors over L. Every algebraic extension F/K admits a normal closure L, which is an extension field of F such that L/K is normal and which is minimal with this property.

An algebraic extension L/K is called separable
Separable extension
In modern algebra, an algebraic field extension E\supseteq F is a separable extension if and only if for every \alpha\in E, the minimal polynomial of \alpha over F is a separable polynomial . Otherwise, the extension is called inseparable...

 if the minimal polynomial
Minimal polynomial (field theory)
In field theory, given a field extension E / F and an element α of E that is an algebraic element over F, the minimal polynomial of α is the monic polynomial p, with coefficients in F, of least degree such that p = 0...

 of every element of L over K is separable
Separable polynomial
In mathematics, two slightly different notions of separable polynomial are used, by different authors.According to the most common one, a polynomial P over a given field K is separable if all its roots are distinct in an algebraic closure of K, that is the number of its distinct roots is equal to...

, i.e., has no repeated roots in an algebraic closure
Algebraic closure
In mathematics, 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 in mathematics....

 over K. A Galois extension
Galois extension
In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions ; one also says that the extension is Galois. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.The definition...

 is a field extension that is both normal and separable.

A consequence of the primitive element theorem
Primitive element theorem
In mathematics, more specifically in the area of modern algebra known as field theory, the primitive element theorem or Artin's theorem on primitive elements is a result characterizing the finite degree field extensions that possess a primitive element...

 states that every finite separable extension has a primitive element (i.e. is simple).

Given any field extension L/K, we can consider its automorphism group Aut(L/K), consisting of all field automorphism
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...

s α: L → L with α(x) = x for all x in K. When the extension is Galois this automorphism group is called the Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...

 of the extension. Extensions whose Galois group is abelian
Abelian group
In abstract algebra, 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 their order . Abelian groups generalize the arithmetic of addition of integers...

 are called 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....

s.

For a given field extension L/K, one is often interested in the intermediate fields F (subfields of L that contain K). The significance of Galois extensions and Galois groups is that they allow a complete description of the intermediate fields: there is a bijection
Bijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...

 between the intermediate fields and the subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...

s of the Galois group, described by the fundamental theorem of Galois theory
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 its most basic form, the theorem asserts that given a field extension E /F which is finite and Galois, there is a one-to-one correspondence between its...

.

Generalizations


Field extensions can be generalized to ring extensions which consist of a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

 and one of its subring
Subring
In mathematics, a subring of R is a subset of a ring, is itself a ring with the restrictions of the binary operations of addition and multiplication of R, and which contains the multiplicative identity of R...

s. A closer non-commutative analog are central simple algebra
Central simple algebra
In ring theory and related areas of mathematics a central simple algebra over a field K is a finite-dimensional associative algebra A, which is simple, and for which the center is exactly K...

s (CSAs) – ring extensions over a field, which are simple algebra (no non-trivial 2-sided ideals, just as for a field) and where the center of the ring is exactly the field. For example, the only finite field extension of the real numbers is the complex numbers, while the quaternions are a central simple algebra over the reals, and all CSAs over the reals are Brauer equivalent to the reals or the quaternions. CSAs can be further generalized to Azumaya algebra
Azumaya algebra
In mathematics, an Azumaya algebra is a generalization of central simple algebras to R-algebras where R need not be a field. Such a notion was introduced in a 1951 paper of Goro Azumaya, for the case where R is a commutative local ring...

s, where the base field is replaced by a commutative local ring
Local ring
In abstract algebra, more particularly 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 varieties or manifolds, or of algebraic number fields examined at a particular place, or...

.

Extension of scalars



Given a field extension, one can "extend scalars
Extension of scalars
In abstract algebra, extension of scalars is a means of producing a module over a ring S from a module over another ring R, given a homomorphism f : R \to S between them...

" on associated algebraic objects. For example, given a real vector space, one can produce a complex vector space via complexification
Complexification
In mathematics, the complexification of a real vector space V is a vector space VC over the complex number field obtained by formally extending scalar multiplication to include multiplication by complex numbers. Any basis for V over the real numbers serves as a basis for VC over the complex...

. In addition to vector spaces, one can perform extension of scalars for associative algebra
Associative algebra
In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...

s over defined over the field, such as polynomials or group algebra
Group algebra
In mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra , such that representations of the algebra are related to representations of the group...

s and the associated group representation
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...

s. Extension of scalars of polynomials is often used implicitly, by just considering the coefficients as being elements of a larger field, but may also be considered more formally. Extension of scalars has numerous applications, as discussed in extension of scalars: applications.

See also

  • Field theory
    Field theory (mathematics)
    Field theory is a branch of mathematics which studies the properties of fields. A field is a mathematical entity for which addition, subtraction, multiplication and division are well-defined....

  • Glossary of field theory
    Glossary of field theory
    Field theory is the branch of mathematics in which fields are studied. This is a glossary of some terms of the subject. -Definition of a field:...

  • Tower of fields
  • Primary extension
    Primary extension
    In field theory, a branch of algebra, primary extension L of K is a field extension such that the algebraic closure K in L is purely inseparable over K....

  • Regular extension