Extension of a field

field extension

A field containing the given field as a subfield. The notation means that is an extension of the field . In this case, is sometimes called an overfield of the field .

Let and be two extensions of a field . An isomorphism of fields is called an isomorphism of extensions (or a -isomorphism of fields) if is the identity on . If an isomorphism of extensions exists, then the extensions are said to be isomorphic. If , is called an automorphism of the extension . The set of all automorphisms of an extension forms a group, . If is a Galois extension, this group is denoted by and is called the Galois group of the field over , or the Galois group of the extension . An extension is called Abelian if its Galois group is Abelian.

An element of the field is called algebraic over if it satisfies some algebraic equation with coefficients in , and transcendental otherwise. For every algebraic element there is a unique polynomial , with leading coefficient equal to 1, that is irreducible in the polynomial ring and satisfies ; any polynomial over having as a root is divisible by . This polynomial is called the minimal polynomial of . An extension is called algebraic if every element of is algebraic over . An extension that is not algebraic is called transcendental. An extension is called normal if it is algebraic and if every irreducible polynomial in having a root in factorizes into linear factors in . The subfield is said to be algebraically closed in if every element of that is algebraic over actually lies in , that is, every element of is transcendental over . A field that is algebraically closed in all its extensions is called an algebraically closed field.

An extension is said to be finitely generated (or an extension of finite type) if there is a finite subset of such that coincides with the smallest subfield containing and . In this case one says that is generated by over . If is generated over by one element , then the extension is called simple or primitive and one writes . A simple algebraic extension is completely determined by the minimal polynomial of . More precisely, if is another simple algebraic extension and , then there is an isomorphism of extensions sending to . Furthermore, for any irreducible polynomial there is a simple algebraic extension with minimal polynomial . It can be constructed as the quotient ring . On the other hand, for any simple transcendental extension there is an isomorphism of extensions , where is the field of rational functions in over . Any extension of finite type can be obtained by performing a finite sequence of simple extensions.

An extension is called finite if is finite-dimensional as a vector space over , and infinite otherwise. The dimension of this vector space is called the degree of and is denoted by . Every finite extension is algebraic and every algebraic extension of finite type is finite. The degree of a simple algebraic extension coincides with the degree of the corresponding minimal polynomial. On the other hand, a simple transcendental extension is infinite.

Suppose one is given a sequence of extensions . Then is algebraic if and only if both and are. Further, is finite if and only if and are, and then

If and are two algebraic extensions and is the compositum of the fields and in a common overfield, then is also algebraic.

See also Separable extension; Transcendental extension.

References

[1]	N. Bourbaki, "Eléments de mathématique. Algèbre" , Masson (1981) pp. Chapt. 4–7
[2]	B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German)
[3]	O. Zariski, P. Samuel, "Commutative algebra" , 1 , Springer (1975)
[4]	S. Lang, "Algebra" , Addison-Wesley (1974)