of the Gaussian integers. We take θ to be the imaginary unit i, with minimal polynomial H(X) = X2 + 1. Since Z[] is the whole ring of integers of Q(), the conductor is the unit ideal, so there are no exceptional primes. For P = (2), we need to work in the field Z/(2)Z, which amounts to factorising the polynomial X2 + 1 modulo 2: Therefore, there is only one prime factor, with inertia degree 1 and ramification index 2, and it is given by The next case is for P = (p) for a prime p ≡ 3 mod 4. For concreteness we will take P = (7). The polynomial X2 + 1 is irreducible modulo 7. Therefore, there is only one prime factor, with inertia degree 2 and ramification index 1, and it is given by The last case is P = (p) for a prime p ≡ 1 mod 4; we will again take P = (13). This time we have the factorisation Therefore, there are two prime factors, both with inertia degree and ramification index 1. They are given by and See also Chebotarev's density theorem References External links Algebraic number theory Galois theory