2007-10-19

1. Return graded homework
2. Summarize comments from students
3. Prove that if [O_K : O] is coprime to p, then ideals of O that divide p are in bijection with ideals of O_K, and have same generators. Give both proofs.
4. Formally define essential discriminant divisors and recall example (with no details)
5. General factorization:
   1. Given a prime p and an order O=Z[a], find the prime factorization of p*O_K.
   2. Two steps: (1) find the primes of O_K that contain p, then (2) find the powers of each that divide p*O_K.
   3. We focus on (1) for the rest of today.
   4. For this, we find a p-maximal order O' that contains O, then find the primes that contain p in O'.
   5. State the Post-Zassenhaus theorem, and give the proof of part of it.
   6. State lemma 5.3.6: computing a p-maximal order.
   7. Describe algorithm to write finite separable algebra as sum of fields.
   8. Describe general algorithm for factoring primes.
6. A toy implementation of prime factorization and/or finding a p-maximal order would be a good student project. To make it much more interesting make code that is general enough to also do something in the context of a functional field of a curve. Hopefully Alyson will do this ?