Field extension
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. 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.
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 L.As, 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 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.If L is an extension of F, which is in turn an extension of K, then F is said to be an intermediate field of L / K.
Given a field extension, 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 .
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 and, the extension is finite if and only if both and 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. One says that K is the field generated by S over K, and that S is a generating set of K over K. When is finite, one writes instead of and one says that K is finitely generated over K. If S consists of a single element s, the extension is called a simple extension and s is called a primitive element of the extension.
An extension field of the form is often said to result from the of S to K.
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 is not finite, the field K 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 , 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 in order to "create" a root for a given polynomial f. Suppose for instance that K does not contain any element x with x2 = −1. Then the polynomial is irreducible in K, 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.
By iterating the above construction, one can construct a splitting field of any polynomial from K. 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 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 of all rational functions in the variable X with coefficients in K; the elements of K are fractions of two polynomials over K, and indeed K is the field of fractions of the polynomial ring K. 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, is an extension field of K.
Algebraic extension
An element x of a field extension 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 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 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 and are finite. Thus is also finite, as well as the sub extensions, and . It follows that, st and 1/s are all algebraic.
An algebraic extension 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 .
Transcendental extension
Given a field extension, 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 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 if and only if there exists a transcendence basis S of L/K such that L = K. 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 = KNormal, separable and Galois extensions
An algebraic extension L/K is called normal if every irreducible polynomial in K 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.
Given any field extension L/K, we can consider its automorphism group Aut, consisting of all field automorphisms α: L → L with α = 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. 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.