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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 DR 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 DR 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 that is normal in the sense that it contains all the roots of the polynomial that defines it).

If DR is odd and in the above notation N is even, then the primes lying over 2 and . 2 + . and . 2 + . (1 - ). 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 p[T]/(P(T)) is isomorphic to p2 or it splits into two p factors. The former case occurs when DR 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 DR is a square modulo p and we obtain two primes Pp and Qp, lying over p; both these have norm p and their product (and intersection) is pR. Let cp be a number between 1 and p - 1 so that cp2 = DR(mod p); then ap = (1 + cp)/2 satisfies the equation modulo p in the DR odd case and ap = (cp)/2 satisfies the equation modulo p in the DR even case. Thus we can pick a solution ap of the equation modulo p in each case and declare that Pp = . p + . ( - ap). The primes Pp and Qp are interchanged by the involution.

Next: 7.2 Naive computation of Up: 7 Quadratic fields Previous: 7 Quadratic fields
Kapil Hari Paranjape 2002-10-20