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 .
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.
Read more about this topic: Degree Of A Field Extension
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