Commutative Algebra

Abstract

Let K be a field, F a subfield. We then say that K is an extension of the field F. We can see K as a vector space over F by associating to a scalar λ ∈ F and a vector x ∈ K the vector λx, which is equal to the usual product of elements λ, x ∈ K. If the dimension dim _F K of this vector space is finite then K is called a finite extension of the field F, and dim _F K is called the degree of this extension (often it is denoted as (K : F) or [K : F]). For instance, the field of complex numbers C has degree 2 over the field of real numbers R and infinite degree (or to be more exact, no finite degree) over the field of rational numbers Q. An important property of finite extensions is expressed by the following lemma.