In mathematics, more specifically in abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties. Euclid, detail from The School of Athens by Raphael. ... Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... Field theory is a branch of mathematics which studies the properties of fields. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

Field extensions can be generalized to ring extension which consists of a ring and one of its subrings. In mathematics, more specifically in ring theory, a ring extension or extension ring is a ring R with a subring S. We write R/S and say R is a ring extension of S Given a ring extension R/S and a two prime ideals P in R and p... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... In abstract algebra, a branch of mathematics, a subring is a subset of a ring containing the multiplicative identity, which is itself a ring under the same binary operations. ...

Definitions

Given two fields K and L, if K is a subset of L and the field operations of addition and multiplication in K are the same as those in L, we say that K is a subfield of L, L is an extension field of K and that L/K, read as "L over K", is a field extension. A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. The relationship of one set being a subset of another is called inclusion. ...

If L is an extension of F which is in turn an extension of K, then we say F is an intermediate field or subextension of the field extension L/K.

Given a field extension L/K and a subset S of L, we denote by K(S) the smallest subfield of L which contains K and S. We say K(S) is generated by the adjunction of elements of S to K. If S consists of only one element s we often write K(s) instead of K({s}). A field extension of the form L=K(s) is called a simple extension and s is called a primitive element of the extension. In mathematics, a primitive element for an extension of fields L/K is an element ζ of L such that L = K(ζ), or in other words such that L is generated by ζ over K. This means that every element of L can be written as a quotient of two polynomials in...

Given a field extension L/K, then L can also be considered as a vector space over K. The elements of L are the "vectors" and the elements of K are the "scalars". We add the vectors just like we add elements in L, and scalar multiplication is multiplication of elements from L by elements from K. The dimension of this vector space is called the degree of the extension, and is denoted by [L : K]. Vector spaces (or linear spaces) are spaces whose elements, known as vectors, can be scaled and added; all linear combinations can be formed. ... In mathematics, the dimension of a vector space V is the cardinality (i. ...

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.

Notes

The notation L/K is purely formal and does not imply the formation of a quotient ring or quotient group or any other kind of division. In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that collapses the normal subgroup N to the identity element. ...

It is often desirable to talk about field extensions in situations where the small field is not actually contained in the larger one, but is naturally embedded. For this purpose, one abstractly defines a field extension as an injective ring homomorphism between two fields. Every ring homomorphism between fields is injective, so field extensions are precisely the morphisms in the category of fields. In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ...

In the sequel, we will suppress the injective homomorphism and assume that we are dealing with actual subfields.

Examples

The field of complex numbers C is an extension field of the field of real numbers R, and R in turn is an extension field of the field of rational numbers 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(i). [R : Q] = c (the cardinality of the continuum), so this extension is infinite. Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... 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, the cardinality of the continuum is the cardinal number of the set of real numbers R (sometimes called the continuum). ...

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 fields and are important in number theory. 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... To meet Wikipedias quality standards, this article or section may require cleanup. ...

Another extension field of the rationals, quite different in flavor, is the field of p-adic numbers Q_p for a prime number p. The p-adic number systems were first described by Kurt Hensel in 1897. ...

It is common to construct an extension field of a given field K as a quotient ring of the polynomial ring 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 in K[X], consequently the ideal (X² + 1) generated by this polynomial is maximal, 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). In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ... In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ... In mathematics, the adjective irreducible means that an object cannot be expressed as a product of at least two non-trivial factors in a given ring. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ... In mathematics, more specifically in ring theory a maximal ideal is a special kind of ideal which is in some sense maximal, that is not contained in any other non-trivial ideal of the ring. ...

By iterating the above construction, one can construct the splitting field 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. In abstract algebra, the splitting field of a polynomial P(X) over a given field K is a field extension L of K, over which P factorizes into linear factors X − ai, and such that the ai generate L over K. It can be shown that such splitting fields exist...

If p is any prime number and n is a positive integer, we have a finite field GF(pⁿ) with pⁿ elements; this is an extension field of the finite field GF(p) = Z/pZ with p elements. In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. ...

Given a field K, we can consider the field K(X) of all rational functions in the variable X with coefficients in K; the elements of K(X) are fractions of two polynomials over K, and indeed K(X) is the field of fractions of the polynomial ring K[X]. This field of rational functions is an extension field of K. We have [K(X) : K] = (aleph-null, the cardinality of countably infinite sets), so this extension is infinite. In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ... In mathematics, a polynomial is an expression in which constants and powers of variables are combined using (only) addition, subtraction, and multiplication. ... In mathematics, every integral domain can be embedded in a field; the smallest field which can be used is the quotient field or the field of fractions of the integral domain. ... In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ... In the branch of mathematics known as set theory, the aleph numbers are a series of numbers used to represent the cardinality (or size) of infinite sets. ... In mathematics, a countable set is a set with the same cardinality (i. ...

Given a Riemann surface M, the set of all meromorphic functions 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 defined on M. In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ... 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. ... In mathematics a constant function is a function whose values do not vary and thus are constant. ...

Given an algebraic variety V over some field K, then the function field of V, consisting of the rational functions defined on V and denoted by K(V), is an extension field of K. In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ... In algebraic geometry, the function field of an algebraic variety V is the field of fractions of the ring of regular functions on V. Since algebraic varieties are irreducible varieties by definition, the ring of regular functions on V is an integral domain, and hence has a field of fractions. ...

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 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 group theory, given a group G under a binary operation *, we say that some subset H of G is 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 is a group...

In particular then, the characteristics of L and K are the same. In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ...

Algebraic and transcendental elements

If L is an extension of K, then an element of L which is a root of a nonzero polynomial over K is said to be algebraic over K. Elements that are not algebraic are called transcendental. As an example: In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ... In mathematics, a polynomial is an expression in which constants and powers of variables are combined using (only) addition, subtraction, and multiplication. ... In mathematics, the roots of polynomials are in abstract algebra called algebraic elements. ... In mathematics, the roots of polynomials are in abstract algebra called algebraic elements. ...

• 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 x³-10x²+2
• In R/Q, e is transcendental because there is no polynomial with rational coefficients that has e as a root (see transcendental number)
• 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 and transcendental number are used to describe the complex numbers that are algebraic and transcendental (respectively) over Q. In mathematics, a transcendental number is any real number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. ... In mathematics, an algebraic number relative to a field is any element of a given field containing such that is a solution of a polynomial equation of the form: anxn + an−1xn−1 + ··· + a1x + a0 = 0 where n is a positive integer called the degree of the polynomial, every coefficient... In mathematics, a transcendental number is any real number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. ...

If every element of L is algebraic over K, then the extension L/K is said to be an algebraic extension; otherwise it is said to be transcendental. If every element of L except those in K is transcendental over K, then the extension is said to be purely transcendental. In abstract algebra, a field extension L /K is called algebraic if every element of L is algebraic over K, i. ... In abstract algebra, the transcendence degree of a field extension L/K is a certain rather coarse measure of the size of the extension. ...

It can be shown that an extension is algebraic if and only if 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; this is essentially the largest extension field of K that is algebraic over K and it contains all roots of all polynomial equations with coefficients in K. For example, C is the algebraic closure of Q (and also of R). In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ...

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 of L/K. Given any algebraically independent set S over K, then K(S)/K is purely transcendental. 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. 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. This means that for every finite sequence α1, ..., αn of elements of S, no two the... In abstract algebra, the transcendence degree of a field extension L / K is a certain rather coarse measure of the size of the extension. ... In abstract algebra, the transcendence degree of a field extension L/K is a certain rather coarse measure of the size of the extension. ...

Normal, separable and Galois extensions

A field extension L/K is called normal if every irreducible polynomial 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. 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]. The following conditions are equivalent to L/K being a normal extension: Let Ka an algebraic closure of K containing L. Every... In mathematics, the adjective irreducible means that an object cannot be expressed as a product of at least two non-trivial factors in a given ring. ... In group theory, the conjugate closure of a subset S of a group G is the subgroup of G which is generated by the elements of S and their conjugates SG = {x ∈ G | there exists g ∈ G and s ∈ S such that x = g−1sg}, The conjugate closure of S...

An algebraic extension L/K is called separable if the minimal polynomial of every element of L over K is separable. A Galois extension is a field extension that is both normal and separable. In mathematics, a separable extension of a field K is a field L containing K that can be generated by adjoining to K a set of elements α, each of which is a root of a separable polynomial over K. In that case, each β in L has a separable... In mathematics, the minimal polynomial of an object α is the monic polynomial p of least degree such that p(α)=0. ... In mathematics, a polynomial P(X) is separable over a field K if its roots in an algebraic closure of K are distinct - that is P(X) has distinct linear factors in some large enough field extension. ... In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions (described below); one also says that the extension is Galois. ...

A consequence of the primitive element theorem states that every finite separable extension has a primitive element (i.e. is simple). In mathematics, a primitive element for an extension of fields L/K is an element ζ of L such that L = K(ζ), or in other words such that L is generated by ζ over K. This means that every element of L can be written as a quotient of...

Given any field extension L/K, we can consider its automorphism group Aut(L/K), consisting of all field automorphisms α : L → L with α(x) = x for all x in K. In case the extension is Galois, this automorphism group is called the Galois group of the extension. Extensions whose Galois group is abelian are called abelian extensions. In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, a Galois group is a group associated with a certain type of field extension. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In abstract algebra, an abelian extension is a field extension for which the associated Galois group is abelian. ...

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 between the intermediate fields and the subgroups of the Galois group, described by the fundamental theorem of Galois theory. A bijective function. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is 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 is a group... In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions. ...

See also

• Field theory
• Glossary of field theory