How to factor a polynomial modulo p?

joebohr

I can understand most of Galois Theory and Number Theory dealing with factorization and extension fields, but I always run into problems that involve factorization mod p, which I can't seem to figure out how to do. I can't find any notes anywhere either, so I was wondering if someone could give me some steps. p is prime, of course.

morphism

Do you have any specific questions? For general reading material, you could try googling "factoring polynomials over finite fields".

joebohr

I seem to have figured out how to factor mod p (in a prime field) between a couple documents:

www.science.unitn.it/~degraaf/compalg/polfact.pdf

http://www.math.uiuc.edu/~r-ash/Ant/AntChapter4.pdf

However, I'm still wondering what other types of finite fields it would be useful to factor over (am I correct in assuming that not all finite fields are prime fields?)

camilus

You usually reduce the polynomial using the small Fermat theorem, x^p [itex]\equiv[/itex] x (mod p) for every variable x that has a power greater than p-1