Integral Dependence and Normal Varieties

b + an−1b + · · ·+ a0 = 0. Example 1.2. Let A = Z and B = Q. Then b ∈ Q is integral over Z if and only if b ∈ Z. Indeed, if b ∈ Z then b solves x − b ∈ Z[x]. Conversely, write b = c/d for relatively prime integers c and d, d > 0. Let f(x) = x + an−1x + · · ·+ a0 be a polynomial with integer coefficients that b satisfies. Substituting b for x and multiplying by d we obtain c = −(dan−1c + · · ·+ da0). Since d divides the right hand side, we get that d|c. But (d, c) = 1. Therefore, d = 1.