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  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 ?

2013-05-11 18:33