We had the following problem for homework: let prove that lies in a cyclotomic extension.

(Without using Kronecker-Weber Theorem!)

I found a solution to this problem using quadradic Gauss sums, however this is not a course in number theory - it is a course on Galois theory, therefore my solution might be not acceptable. I will post it anyway because I want to print it out and give it to the class so which is why I am hanging it on the forum. I am interested in seeing other solutions.