# User Contributed Dictionary

### Noun

# Extensive Definition

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.

Field extensions can be generalized to ring
extension which consists of a ring
and one of its subrings.

## Definitions

Let L be a field. If K is a subset of L which is closed with respect to the field operations of addition and multiplication in L and the additive and multiplicative inverses of every element in K are in K, 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.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(). A field extension of the form L=K(s) is
called a simple
extension 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
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].

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 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 ring homomorphism
between fields is injective, so field extensions are precisely the
morphisms in the
category
of fields.

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 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.The set Q(√2) = is an extension field
of Q, also clearly a simple extension. The degree is 2 because can
serve as a basis. Finite extensions of Q are also called algebraic
number fields and are important in number
theory.

Another extension field of the rationals, quite
different in flavor, is the field of p-adic
numbers Qp 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 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).

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.

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.

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 C(M). It is an
extension field of C, if we identify every complex number with the
corresponding constant
function defined on M.

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.

## 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−,·) is a subgroup of (L−,·). 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
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 over K is said to be algebraic 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)
- In C/R, e is algebraic because it is the root of x−e

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.

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 R.

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.

## 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.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 L. 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 : L → L
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.

## See also

subfield in German: Körpererweiterung

subfield in Spanish: Extensión de cuerpo

subfield in French: Extension de corps

subfield in Korean: 체의 확대

subfield in Italian: Estensione di campi

subfield in Hebrew: הרחבת שדות

subfield in Japanese: 体の拡大

subfield in Polish: Rozszerzenie ciała

subfield in Russian: Конечное расширение

subfield in Finnish:
Kuntalaajennus