Past exam of the options I course of the University of Oxford / 2004 / B / B1 / i / Solution

The inclusion $F\subseteq K$ makes $K$ a vector space over $F$ by its own addition and multiplication by elements of $F$. The extension is finite when this vector space has a finite basis. Its degree of a field extension isBeing a finite field extension concerns vector-space dimension, not the cardinality of the underlying field.