To complete the proof, we would show that if , then is already -maximal. See [Coh93, §6.1.1] for the rest if this proof.
After deciding on how to represent elements of and orders and ideals in , one can give an efficient algorithm to compute the of the theorem. The algorithm mainly involves linear algebra over finite fields. It is complicated to describe, but efficient in practice, and is conceptually simple--just compute . The trick for reducing the computation of to linear algebra is the following lemma:
Note that to give an algorithm one must also figure out how to explicitly compute and the kernel of this map (see the next section for more details).
William Stein 2012-09-24