Bump!
Apparently, if the extension is finite separable, which the above extension is, then the trace is nonzero. What is wrong here?
Let [; a = \sqrt{2}+i \sqrt{3} ;] and [; K = Q(a) ;]. Compute the norm [; N_{K/Q}(a) ;] and the trace [; Tr_{K/Q}(a) ;].
Well, the minimal polynomial of a is x^4+2x^2+25, and according to Lang VI, §5, thm 5.1, the norm is (-1)^4 25 = 25 and the trace -a_{n-1} = -0 = 0. Easy. Am I missing something or is this a "did you even read the book" type of question?