Field extension

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In mathematics, particularly in algebra, a field extension is a pair of fields such that the operations of E are those of F restricted to E. In this case, F is an extension field of E and E is a subfield of F. [1] [2] [3] For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers.

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Field extensions are fundamental in algebraic number theory, and in the study of polynomial roots through Galois theory, and are widely used in algebraic geometry.

Subfield

A subfield of a field L is a subset K of L that is a field with respect to the field operations inherited from L. Equivalently, a subfield is a subset that contains 1, and is closed under the operations of addition, subtraction, multiplication, and taking the inverse of a nonzero element of K.

As 1 – 1 = 0, the latter definition implies K and L have the same zero element.

For example, the field of rational numbers is a subfield of the real numbers, which is itself a subfield of the complex numbers. More generally, the field of rational numbers is (or is isomorphic to) a subfield of any field of characteristic 0.

The characteristic of a subfield is the same as the characteristic of the larger field.

Extension field

If K is a subfield of L, then L is an extension field or simply extension of K, and this pair of fields is a field extension. Such a field extension is denoted L / K (read as "L over K").

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 L / K.

Given a field extension L / K, the larger field L is a K-vector space. The dimension of this vector space is called the degree of the extension and is denoted by [L : K].

The degree of an extension is 1 if and only if the two fields are equal. In this case, the extension is a trivial extension. Extensions of degree 2 and 3 are called quadratic extensions and cubic extensions, respectively. A finite extension is an extension that has a finite degree.

Given two extensions L / K and M / L, the extension M / K is finite if and only if both L / K and M / L are finite. In this case, one has

Given a field extension L / K and a subset S of L, there is a smallest subfield of L that contains K and S. It is the intersection of all subfields of L that contain K and S, and is denoted by K(S). One says that K(S) is the field generated by S over K, and that S is a generating set of K(S) over K. When is finite, one writes instead of and one says that K(S) is finitely generated over K. If S consists of a single element s, the extension K(s) / K is called a simple extension [4] [5] and s is called a primitive element of the extension. [6]

An extension field of the form K(S) is often said to result from the adjunction of S to K. [7] [8]

In characteristic 0, every finite extension is a simple extension. This is the primitive element theorem, which does not hold true for fields of non-zero characteristic.

If a simple extension K(s) / K is not finite, the field K(s) is isomorphic to the field of rational fractions in s over K.

Caveats

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. Instead the slash expresses the word "over". In some literature the notation L:K is used.

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 non-zero ring homomorphism between fields is injective because fields do not possess nontrivial proper ideals, so field extensions are precisely the morphisms in the category of fields.

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

Examples

The field of complex numbers is an extension field of the field of real numbers , and in turn is an extension field of the field of rational numbers . Clearly then, is also a field extension. We have because is a basis, so the extension is finite. This is a simple extension because (the cardinality of the continuum), so this extension is infinite.

The field

is an extension field of also clearly a simple extension. The degree is 2 because can serve as a basis.

The field

is an extension field of both and of degree 2 and 4 respectively. It is also a simple extension, as one can show that

Finite extensions of are also called algebraic number fields and are important in number theory. Another extension field of the rationals, which is also important in number theory, although not a finite extension, is the field of p-adic numbers for a prime number p.

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 is irreducible in K[X], consequently the ideal generated by this polynomial is maximal, and 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 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 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 with p 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. This extension is infinite.

Given a Riemann surface M, the set of all meromorphic functions defined on M is a field, denoted by It is a transcendental extension field of if we identify every complex number with the corresponding constant function defined on M. More generally, 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.

Algebraic extension

An element x of a field extension L / K is algebraic over K if it is a root of a nonzero polynomial with coefficients in K. For example, is algebraic over the rational numbers, because it is a root of If an element x of L is algebraic over K, the monic polynomial of lowest degree that has x as a root is called the minimal polynomial of x. This minimal polynomial is irreducible over K.

An element s of L is algebraic over K if and only if the simple extension K(s) /K is a finite extension. In this case the degree of the extension equals the degree of the minimal polynomial, and a basis of the K-vector space K(s) consists of where d is the degree of the minimal polynomial.

The set of the elements of L that are algebraic over K form a subextension, which is called the algebraic closure of K in L. This results from the preceding characterization: if s and t are algebraic, the extensions K(s) /K and K(s)(t) /K(s) are finite. Thus K(s, t) /K is also finite, as well as the sub extensions K(s ± t) /K, K(st) /K and K(1/s) /K (if s ≠ 0). It follows that s ± t, st and 1/s are all algebraic.

An algebraic extensionL / K is an extension such that every element of L is algebraic over K. Equivalently, an algebraic extension is an extension that is generated by algebraic elements. For example, is an algebraic extension of , because and are algebraic over

A simple extension is algebraic if and only if it is finite. This implies that an extension is algebraic if and only if it is the union of its finite subextensions, and that every finite extension is algebraic.

Every field K has an algebraic closure, which is up to an isomorphism the largest extension field of K which is algebraic over K, and also the smallest extension field such that every polynomial with coefficients in K has a root in it. For example, is an algebraic closure of , but not an algebraic closure of , as it is not algebraic over (for example π is not algebraic over ).

Transcendental extension

Given a field extension L / K, 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. 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—a class of such extensions take the form L/K where both L and K are algebraically closed. 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 where x is transcendental over The set is algebraically independent since x is transcendental. Obviously, the extension is algebraic, hence is a transcendence basis. It doesn't generate the whole extension because there is no polynomial expression in for . But it is easy to see that is a transcendence basis that generates so this extension is indeed purely transcendental.)

Normal, separable and Galois extensions

An algebraic 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.

An algebraic extension L/K is called separable if the minimal polynomial of every element of L over K is separable, i.e., has no repeated roots in an algebraic closure over K. A Galois extension is a field extension that is both normal and separable.

A consequence of the primitive element theorem 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 automorphisms α: LL with α(x) = x for all x in K. When 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.

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.

Generalizations

Field extensions can be generalized to ring extensions which consist of a ring and one of its subrings. A closer non-commutative analog are central simple algebras (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 algebras, where the base field is replaced by a commutative local ring.

Extension of scalars

Given a field extension, one can "extend scalars" on associated algebraic objects. For example, given a real vector space, one can produce a complex vector space via complexification. In addition to vector spaces, one can perform extension of scalars for associative algebras defined over the field, such as polynomials or group algebras and the associated group representations. 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

Notes

  1. Fraleigh (1976 , p. 293)
  2. Herstein (1964 , p. 167)
  3. McCoy (1968 , p. 116)
  4. Fraleigh (1976 , p. 298)
  5. Herstein (1964 , p. 193)
  6. Fraleigh (1976 , p. 363)
  7. Fraleigh (1976 , p. 319)
  8. Herstein (1964 , p. 169)

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Constructible number Real number constructible via compass and straightedge

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Field (mathematics) Algebraic structure with addition, multiplication and division

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 do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.

In mathematics, a finite field or Galois field is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod p when p is a prime number.

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. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them.

In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the field or ring to which the coefficients of the polynomial and its possible factors are supposed to belong. For example, the polynomial x2 − 2 is a polynomial with integer coefficients, but, as every integer is also a real number, it is also a polynomial with real coefficients. It is irreducible if it is considered as a polynomial with integer coefficients, but it factors as if it is considered as a polynomial with real coefficients. One says that the polynomial x2 − 2 is irreducible over the integers but not over the reals.

In abstract algebra, a subset of a field is algebraically independent over a subfield if the elements of do not satisfy any non-trivial polynomial equation with coefficients in .

Field theory is the branch of mathematics in which fields are studied. This is a glossary of some terms of the subject.

In field theory, a subfield of algebra, an algebraic field extension is called a separable extension if for every , the minimal polynomial of over F is a separable polynomial. There is also a more general definition that applies when E is not necessarily algebraic over F. An extension that is not separable is said to be inseparable.

In abstract algebra, the theory of fields lacks a direct product: the direct product of two fields, considered as a ring, is never itself a field. Nonetheless, it is often required to "join" two fields K and L, either in cases where K and L are given as subfields of a larger field M, or when K and L are both field extensions of a smaller field N.

In field theory, the primitive element theorem is a result characterizing the finite degree field extensions that can be generated by a single element. Such a generating element is called a primitive element of the field extension, and the extension is called a simple extension in this case. The theorem states that a finite extension is simple if and only if there are only finitely many intermediate fields. An older result, also often called "primitive element theorem", states that every finite separable extension is simple; it can be seen as a consequence of the former theorem. These theorems imply in particular that all algebraic number fields over the rational numbers, and all extensions in which both fields are finite, are simple.

In mathematics, the (field) norm is a particular mapping defined in field theory, which maps elements of a larger field into a subfield.

In mathematics, 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.

In algebra, a field k is perfect if any one of the following equivalent conditions holds:

In Galois theory, the inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers . This problem, first posed in the early 19th century, is unsolved.

In abstract algebra, a normal extension is an algebraic field extension L/K for which every polynomial that is irreducible over K either has no root in L or splits into linear factors in L. Bourbaki calls such an extension a quasi-Galois extension.

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. It was proved by Évariste Galois in his development of Galois theory.

In field theory, a branch of mathematics, the minimal polynomial of a value α is, roughly speaking, the polynomial of lowest degree having coefficients of a specified type, such that α is a root of the polynomial. If the minimal polynomial of α exists, it is unique. The coefficient of the highest-degree term in the polynomial is required to be 1, and the specified type for the remaining coefficients could be integers, rational numbers, real numbers, or others.

In mathematics, a CM-field is a particular type of number field, so named for a close connection to the theory of complex multiplication. Another name used is J-field.

In mathematics, an algebraic number field is a finite degree field extension of the field of rational numbers . Thus is a field that contains and has finite dimension when considered as a vector space over .

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