Discriminants and Ramified Primes

has some ei greater than 1. If every ei equals 1, we say p is unramified in K. Example 1.1. In Z[i], the only prime which ramifies is 2: (2) = (1 + i)2. Example 1.2. Let K = Q(α) where α is a root of f(X) = T 3 − 9T − 6. Then 6 = α3 − 9α = α(α− 3)(α+ 3). For m ∈ Z, α+m has minimal polynomial f(T −m) in Q[T ], so NK/Q(α+m) = −f(−m) = m3 − 9m+ 6 and the principal ideal (α−m) has norm N(α−m) = |m − 9m+ 6|. Therefore N(α) = 6, N(α − 3) = 6, and N(α + 3) = 6. It follows that (α) = p2p3, (α− 3) = p2p3, and (α+ 3) = p2p3 (so, in particular, α+ 3 and α− 3 are unit multiples of each other). Thus (2)(3) = (6) = (α)(α− 3)(α+ 3) = p2p 2 p3,