Stephen Silverman
1. The proof consists of the observation that Tr(AB)= Tr(BA), where A = [aij] and B = [bji]. Note that if the field had nonzero characteristic then the argument would merely establish that the sizes of the two bases differed by a multiple of the characteristic. 2. The same argument can be used in an obvious way to define the degree of a finite extension of a field of characteristic zero. This leads directly to a proof of the double extension theorem: if E is a field of characteristic zero, F is a finite extension of E and G is a finite extension of F, then G is a finite extension of E and [G: E] = [G: F][F: E]. This approach can be used to demonstrate the impossibility of the duplication of the cube and the trisection of the angle using straightedge and compass to students who have no knowledge of linear algebra.