Degree of a field extension

In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. The concept plays an important role in many parts of mathematics, including algebra and number theory — indeed in any area where fields appear prominently.

Definition and notation

Suppose that E/F is a field extension. Then E may be considered as a vector space over F (the field of scalars). The dimension of this vector space is called the degree of the field extension, and it is denoted by [E:F].

The degree may be finite or infinite, the field being called a finite extension or infinite extension accordingly. An extension E/F is also sometimes said to be simply finite if it is a finite extension; this should not be confused with the fields themselves being finite fields (fields with finitely many elements).

The degree should not be confused with the transcendence degree of a field; for example, the field Q(X) of rational functions has infinite degree over Q, but transcendence degree only equal to 1.