How to factor a polynomial modulo p?

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This discussion focuses on the process of factoring polynomials modulo a prime number p, specifically within the context of Galois Theory and Number Theory. Key resources identified include the documents "www.science.unitn.it/~degraaf/compalg/polfact.pdf" and "http://www.math.uiuc.edu/~r-ash/Ant/AntChapter4.pdf" which provide insights into polynomial factorization in prime fields. The small Fermat theorem, which states that \( x^p \equiv x \mod p \) for any variable x with a power greater than p-1, is crucial for reducing polynomials in this context. The discussion also raises questions about the applicability of factorization techniques to other types of finite fields beyond prime fields.

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joebohr
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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.
 
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Do you have any specific questions? For general reading material, you could try googling "factoring polynomials over finite fields".
 
morphism said:
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?)
 
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You usually reduce the polynomial using the small Fermat theorem, xp \equiv x (mod p) for every variable x that has a power greater than p-1
 

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