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I have a polynomial, let's say $p(x)=x^5+x^4+x^2+x+2$ (or any other polynomial with rational coefficients). What is the general recommended way of factoring it into irreducible factors (in $Z[x]$, $Q[x]$, $Z_n[x]$, $F_n$, etc.)?

I've tried to use Horner's method on $p(x)$, but it does not have integer roots. I've studied the theory around finite fields for a few days, but I still don't understand how can I factor polynomials or show that the polynomial is irreducible in a given field. Can you possibly recommend me some good website/book with examples how to approach this kind of problems?

dash
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    Factoring by hand is hard but there are algorithms. See Wikipedia. See also http://math.stackexchange.com/questions/26135/is-factoring-polynomials-as-hard-as-factoring-integers. For your example, you can ask WA. – lhf Feb 17 '16 at 10:54

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Thr best currently known algorithm is the one due to Mark van Hoeij, described in this nice paper by Kluners.

Igor Rivin
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  • Thank you, but I've just started with finite fields this semester, so I'm not looking for the best currently known algorithm. I'm looking for a simple way which can be used in an exam. – dash Feb 17 '16 at 11:24
  • Unfortunately, there is no method I know that is both general and easy to use by hand... – Igor Rivin Feb 17 '16 at 11:25