# 2007-10-19

- Return graded homework
- Summarize comments from students
- 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.
- Formally define essential discriminant divisors and recall example (with no details)
- General factorization:
- Given a prime p and an order O=Z[a], find the prime factorization of p*O_K.
- Two steps: (1) find the primes of O_K that contain p, then (2) find the powers of each that divide p*O_K.
- We focus on (1) for the rest of today.
- For this, we find a p-maximal order O' that contains O, then find the primes that contain p in O'.
- State the Post-Zassenhaus theorem, and give the proof of part of it.
- State lemma 5.3.6: computing a p-maximal order.
- Describe algorithm to write finite separable algebra as sum of fields.
- Describe general algorithm for factoring primes.

- 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 ?