How to factor a polynomial modulo p?

  • Context: Graduate 
  • Thread starter Thread starter joebohr
  • Start date Start date
  • Tags Tags
    Polynomial
Click For Summary

Discussion Overview

The discussion revolves around the factorization of polynomials modulo a prime number p. Participants explore concepts related to factorization in finite fields, particularly focusing on methods and resources for understanding this topic better.

Discussion Character

  • Exploratory, Technical explanation, Homework-related

Main Points Raised

  • One participant expresses difficulty in understanding factorization mod p despite familiarity with Galois Theory and Number Theory.
  • Another participant suggests looking for general reading material on factoring polynomials over finite fields and provides links to specific documents that may help.
  • A participant questions the applicability of factorization methods to other types of finite fields, indicating uncertainty about the relationship between prime fields and other finite fields.
  • One participant mentions using the small Fermat theorem to reduce polynomials, stating that for any variable x with a power greater than p-1, xp is congruent to x modulo p.

Areas of Agreement / Disagreement

Participants do not reach a consensus on specific methods for factorization, and multiple viewpoints regarding the applicability of techniques to different types of finite fields remain. The discussion is unresolved regarding the best approach to factor polynomials mod p.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about finite fields and the specific steps involved in the factorization process, which are not fully detailed or agreed upon.

joebohr
Messages
57
Reaction score
0
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.
 
Physics news on Phys.org
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?)
 
Last edited by a moderator:
You usually reduce the polynomial using the small Fermat theorem, xp [itex]\equiv[/itex] x (mod p) for every variable x that has a power greater than p-1
 

Similar threads

Undergrad Quintic Polynomial Tschirnhaus Transform: Possible complex Bring Radicals
  • · Replies 0 ·
Replies
0
Views
3K
Undergrad How to show ##p(x)=g(x)x\pm 1\in\Bbb{Q}[x]## is irreducible in ##\Bbb{Q}_{\Bbb{Z}}[x]##?
  • · Replies 48 ·
2
Replies
48
Views
6K
MHB The extension is Galois iff E is a splitting field of a separable polynomial of F[x]
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Don't understand proof of uniqueness theorem for polynom factorization
  • · Replies 8 ·
Replies
8
Views
2K
MHB The polynomial is irreducible over Q(i)
  • · Replies 16 ·
Replies
16
Views
4K
Undergrad ##(A/\mathfrak{a})_{\mathfrak{p}/\mathfrak{a}}## and its isomorphism?
  • · Replies 31 ·
2
Replies
31
Views
3K
Undergrad Generators of Galois group of ## X^n - \theta ##
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Splitting ring of polynomials - why is this result unfindable?
  • · Replies 9 ·
Replies
9
Views
2K
Graduate Computing the Galois Group of a Univariate Polynomial (Irreducible or reducible)
  • · Replies 23 ·
Replies
23
Views
6K
Undergrad Separable Polynomials - Remarks by Dummit and Foote .... ....
  • · Replies 1 ·
Replies
1
Views
2K