Let be an algebraic extension of of degree , and let be the ring of integers in . Let . Show that the norm is precisely . (The first norm is the norm of as an ideal, while the second norm is the norm of as an element.)
We can assume without loss of generality that is positive.
Recall that the conjugates of over are all the same (Theorem 5.10 in TAN), so that .
Let be an integral basis for . Suppose is an ideal containing . By Theorem 9.3 in TAN, there exists an element such that . Note that we may assume that each is in . In particular, . We claim that this is in fact unique. To that end, suppose mod for some . Then we have for some . Since is an integral basis, we have mod for each , and thus . So in fact , as desired.