§ Scatted algebraic number theory ideas: Ramification
- I've had Pollion on math IRC explain ramification to me.
15:17 Take your favorite dedekind domain.
15:17 For instance, consider K a number field
15:17 and O_K the ring of integers.
15:17 Then take a prime p in Z.
15:18 Since Z \subset O_K, p can be considered as an element of O_K, right ?
15:18 Ok. p is prime in Z, meaning that the ideal (p) = pZ is a prime ideal of Z.
15:18 Consider now this ideal, but in O_K
15:19 ie the ideal pO_K
15:19 It may not be prime anymore
15:19 So it factors as a product of prime ideals *of O_K*
15:20 pO_K = P_1^e_1....P_r^e_r
15:20 where P_i are distinct prime ideals of O_K.
15:20 You say that p ramifies in O_K (or in K) when there is some e_i which is > 1
15:21 Take Z[i], the ring of Gauss integers.
15:22 It is the ring of integers of the field Q(i).
15:22 Take the prime 2 in Z.
15:23 (2) = (1 + i) (1 - i) in Z[i] ?
15:23 But in fact
15:23 The ideal (1-i) = (1+i) (as ideals)
15:23 So (2) = (1+i)^2
15:23 And you can prove that (1+i) is a prime ideal in Z[i]
15:23 is it because (1 - i)i = i + 1 = 1 + i?
15:24 very cool
15:24 Therefore, (2) ramifies in Z[i].
15:24 is it prime because the quotient Z[i]/(1 - i) ~= Z is an integral domain? [the quotient tells us to make 1 - i = 0, or to set i = ]
15:24 But you can also prove that primes that ramify are not really common
15:24 it = (1 - i)
15:25 In fact, 2 is the *only* prime that ramifies in Z[i]
15:25 More generally, you only have a finite number of primes that ramify
15:25 in any O_K?