Let be an algebraic element of degree over . Show that is algebraic over degree at most over , and that is algebraic of degree at most over . In each case, show by an example that both “<” and “=” are possible.
By Theorem 4.9 in TAN, since (clearly) the degree of is at most . Now suppose the minimal polynomial of over is ; then certainly is a root of , which has degree . Since the minimal polynomial of divides , the degree of is at most .
Let . Since , is algebraic over of degree 1. is also algebraic of degree 1, so that . However, has degree 2, so that .
On the other hand, if , then , and .
Now consider ; this element is algebraic of degree 2, while has degree 1. So .