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Consider the polynomial $f(x)= x^4-x^3+14x^ 2 + 5x+16$. Also for a prime $p$, let $\mathbb F_p$ denote the field with $p$ elements. Which of the following are always true?

  1. Considering $f$ as a polynomial with coefficients in $\mathbb F_3$, it is a product of two irreducible factors of degree 2 over $\mathbb F_3$.

  2. Considering $f$ as a polynomial with coefficients in $\mathbb F_7$, it has an irreducible factor of degree 3 over $\mathbb F_7$.

  3. $f$ is a product of two polynomials of degree 2 over $\mathbb Z$.

I don't know how to factorise a polynomial over $\mathbb F_p$. Is there any algorithm to do so?

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Hint $\rm\ mod\ 7\!:\ f(x) \equiv x^4\!-x^3\!-2x+2 \equiv (x-1)(x^3\!-2),\: $ and $\rm\:a^3\!\equiv 2\:\Rightarrow 1\equiv a^6\equiv 4\:\Rightarrow\Leftarrow$

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1) False as $x=2$ is root, so not possible to write as product of irreducible factors of degree $2$.

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This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review – Leucippus Mar 2 at 17:51

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