Let be a field, and let be algebraic over . Prove that if is odd, then .
The inclusion is immediate.
Let be the minimal polynomial of over . Now write , separating the even and odd terms. Now , and we have . Since has odd degree and the powers of are linearly independent over , . Thus , and we have .
Note that our proof only needs some nonzero odd-degree term in the minimal polynomial of , so in some cases the proof holds if has even degree.