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 .
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