7.1 Prime ideals

By the earlier analysis, we see that every prime ideal is either ramified (to order 2) or of degree 1 or of degree 2. If the prime lies over 2 then it is not ramified when D_R is odd since, in that case the equation takes the form T² + T or T² + T + 1 modulo 2; both these equations have distinct roots. When D_R is even, the prime over 2 is ramified. When the prime lies over an odd prime p, it is clear that the prime is ramified when the discriminant is divisible by p. Thus the ramified primes are precisely those that lie over primes p that divide the discriminant. (This is also true for any field extension of $\mathbb {Q}$ that is normal in the sense that it contains all the roots of the polynomial that defines it).

If D_R is odd and in the above notation N is even, then the primes lying over 2 and $\mathbb {Z}$ ^. 2 + $\mathbb {Z}$ ^. $\alpha$ and $\mathbb {Z}$ ^. 2 + $\mathbb {Z}$ ^. (1 - $\alpha$). When N is odd, then the only prime lying over 2 is 2R. Now, if p is an odd prime that does not divide the discriminant then either $\mathbb {F}$_p[T]/(P(T)) is isomorphic to $\mathbb {F}$_p² or it splits into two $\mathbb {F}$_p factors. The former case occurs when D_R is not a square modulo p and in this case the prime lying over p is just the ideal pR; which is principal. In the second case D_R is a square modulo p and we obtain two primes P_p and Q_p, lying over p; both these have norm p and their product (and intersection) is pR. Let c_p be a number between 1 and p - 1 so that c_p² = D_R(mod p); then a_p = (1 + c_p)/2 satisfies the equation modulo p in the D_R odd case and a_p = (c_p)/2 satisfies the equation modulo p in the D_R even case. Thus we can pick a solution a_p of the equation modulo p in each case and declare that P_p = $\mathbb {Z}$ ^. p + $\mathbb {Z}$ ^. ($\alpha$ - a_p). The primes P_p and Q_p are interchanged by the involution.