# Field extension

### From Exampleproblems

In abstract algebra, an **extension** of a field *K* is a field *L* which contains *K* as a subfield.
For example, **C** (the field of complex numbers)
is an extension of **R** (the field of real numbers),
and **R** is itself an extension of **Q**
(the field of rational numbers).
The notation *L*/*K* is often used to denote the fact that *L* is an extension of *K*.

More generally, an extension of *K* is a separate pair of fields *K** and *L*, where *L* contains *K** as a subfield, and *K* is isomorphic to *K**. Where it does not cause confusion, we identify *K* and *K**, as above and below.

Given a field extension *L*/*K*,
*L* can be considered as a vector space over *K*,
with vector addition being the field addition on *L*,
and scalar multiplication being a restriction of the field multiplication on *L*.
The dimension of this vector space
is called the **degree** of the extension, and is denoted [*L* : *K*].
The extension is said to be **finite** or **infinite** according as
the degree is finite or infinite.
For example, [**C** : **R**] = 2, so this extension is finite.
By contrast, [**R** : **Q**] = *c* (the cardinality of the continuum),
so this extension is infinite.
If *M* is an extension of *L* which is an extension of *K*,
then [*M* : *K*] = [*M* : *L*].[*L* : *K*].

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*.
If it is not algebraic then it is said to be *transcendental*.
(The special case where *L* = **C** and *K* = **Q** is particularly important. see algebraic number and transcendental number.)
If every element of *L* is algebraic over *K*,
then the extension *L*/*K* is said to be **algebraic**,
otherwise it is said to be **transcendental**.
If every element of *L* \ *K* is transcendental over *K*,
then the extension is said to be **pure 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**, being finite, is algebraic.
But **R**/**Q** is transcendental, although not pure transcendental.
See algebraic extension for more information on algebraic extensions.

If *L*/*K* is a field extension and *V* is a subset of *L*, then the field *K*(*V*) is defined to be the smallest subfield of *L* which contains *K* and *V*. It consists of all those elements of *L* which can be gotten using a finite number of field operations +, -, *, / applied to elements from *K* and *V*. If *L* = *K*(*V*), we say that *L* is **generated** by *V*.

A field extension generated by a single element is called a **simple** extension.
A simple extension is finite if generated by an algebraic element,
and pure transcendental if generated by a transcendental element.
For example, **C** is a simple extension of **R**, as it is generated by *i* (the square root of minus one).
The extension **R**/**Q** is not simple, as it is neither finite nor pure transcendental.

A field extension which has a Galois group is called a Galois extension.
If the Galois group is Abelian, then the extension is called an **Abelian extension**.
For example, **C**/**R** is a Abelian extension, its Galois group being of order 2.
But **R**/**Q** is not a Galois extension, since, for example, the polynomial *x*^{3} − 2, while having a root in **R**, does not split over **R**.de:Körpererweiterung