# Field extension

If and are fields and , then is said to be a **field extension**. We sometimes say that is a field extension of .

If is a field extension, then may be thought of as a vector space over . The dimension of this vector space is called the *degree* of the extension, and is denoted by .

Given three fields , then, if the degrees of the extensions , and , are finite, then are related by the tower law:

One common way to construct an extension of a given field is to consider an irreducible polynomial in the polynomial ring , and then to form the quotient ring . Since is irreducible, is a maximal ideal and so is actually a field. We can embed into this field by , and so we can view as an extension of . Now if we define as , then we can show that in , , and every element of can be expressed as a polynomial in . We can thus think of as the field obtained by 'adding' a root of to .

It can be shown that .

As an example of this, we can now *define* the complex numbers, by .

*This article is a stub. Help us out by expanding it.*