What is the degree of a field extension? What does it mean to be a finite extension?
Solution
For a field extension \(F/K\), addition in \(F\) and multiplication by elements of \(K\) make \(F\) a vector space over \(K\). The degree of \(F\) over \(K\) is \([F:K]\coloneqq\dim_KF\), and \(F/K\) is a finite extension if \([F:K]<\infty\). For example, \([\CC:\RR]=2\), with basis \(1,i\).