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What should I start with when I want to factor a polynomial over $GF(n)$?

Should I start by finding it's irreducible polynomials? Say, I have $x^4+3x^3+4x+1$ and I want to factor it over $GF(5)$ What should I start with when I see a problem like "factor the $a_kx^k*\dots *a_0 $ over $GF(n)$"?

user547075
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user550281
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  • Check for roots. – sharding4 Apr 11 '18 at 00:08
  • See here. – Count Iblis Apr 11 '18 at 00:31
  • By Maple: Consider $\alpha$ be a root of ${\bf f}=x^4+3x^3+4x+1$ then the polynomial $\bf f$ over $GF(5)$ are factored as follows: $${\bf f}= \left( x+4,\alpha \right) \left( x+3,{\alpha}^{3}+4,\alpha+4 \right) \left( x+4,{\alpha}^{3}+{\alpha}^{2}+\alpha+1 \right) \left( x+3,{\alpha}^{3}+4,{\alpha}^{2}+\alpha \right) $$ Commands: 1)alias(alpha = RootOf(x^4+3x^2+4x+1)); 2)Factor(x^4+3x^2+4x+1, alpha) mod 5; Maybe helpful. ( Notice that $4\alpha =-\alpha$) – Amin235 Apr 12 '18 at 20:56

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