Degree of Field Extension
Given that an extension field \(\mathbb{K}\) of \(\mathbb{F}\) can be viewed as a vector space over the subfield, we define the degree of the extension to be the dimension of this vector space.
In the case of simple extensions this is just the degree of the minimal polynomial of the generating element, hence the use of the same word to describe these ideas.
A field extension is called finite if its degree is finite. That is, the extension field has a finite basis over the subfield.
Note: This notion of a finite extension is independent of if the fields are finite as sets. It is of course possible for an extension of an infinite field to be finite, and for that extension field to be infinite. That is, we describe the extension itself as finite. This note is made to draw attention to the fact that terms like finite field extension are somewhat ambiguous as to if they should be read as (finite field) extension or finite (field extension)
A field extension is called infinite if its degree is infinite. That is, the extension field has no finite basis over the subfield.