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]

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

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 $\equiv$ x (mod p) for every variable x that has a power greater than p-1

 

[blue_raver22]

There are several methods for factoring a polynomial modulo p, also known as polynomial reduction modulo p. One approach is to use the fact that for any nonzero integer a, the polynomial f(x) is congruent to f(x+a) modulo p. This means that if we can find an a such that f(x+a) is a simpler polynomial to factor, then we can use this congruence to factor f(x).

Another method is to use the properties of modular arithmetic and the Chinese Remainder Theorem. This theorem states that if we can factor a polynomial modulo p into irreducible polynomials, then we can use these factors to factor the polynomial modulo any other prime number q. This approach involves finding the roots of the polynomial modulo p and then using these roots to construct the factors.

In some cases, it may also be helpful to use the concept of primitive roots. A primitive root of a prime number p is an integer g that generates all of the nonzero elements in the field modulo p. This means that every nonzero element in the field can be written as g^k for some integer k. With this knowledge, we can use the properties of primitive roots to find the roots of a polynomial modulo p and then use these roots to factor the polynomial.

Overall, factoring a polynomial modulo p can be a complex and challenging task, but with a solid understanding of Galois Theory and Number Theory, as well as the use of various techniques such as congruences, the Chinese Remainder Theorem, and primitive roots, it is possible to successfully factor polynomials modulo p. I suggest consulting textbooks or online resources for more detailed explanations and examples of these methods.