Let be a basis for an algebraic number field consisting of algebraic integers such that is squarefree. (Recall that since consists of algebraic integers, is a rational integer.) Prove that is an integral basis.
Let be an integral basis for . ( exists by Theorem 6.9 in TAN.) Since each is an algebraic integer, there exist rational integers such that for each .
Let be the conjugates of and let and . Now , so that certainly , where denotes the th conjugate of and . In particular, we have , where . So . Certainly is a rational integer.
Since is squarefree, in fact , and so . Since is an integral basis, it has minimal discriminant among the bases of (by Theorems 6.9 and 6.10 in TAN), as does . So is an integral basis.