efficient algorithms for factoring polynomials
Do you know of Berlekamp's algorithm and the  Cantor–Zassenhaus algorithm?

http://en.wikipedia.org/wiki/Berlekamp%27s_algorithm

http://en.wikipedia.org/wiki/Cantor%E2%80%93Zassenhaus_algorithm

Unless I'm misunderstanding your question, there is no field of integers- they form a ring but don't have multiplicative inverses