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Let's consider the polynomial $ f\left( x \right) = \left( {x^2 + 2} \right)\prod\limits_{i = - k}^k {\left( {x - 2i} \right) + 2 \in {\Bbb Q}\left[ x \right]} $ . Let's suppose that $ p = 2k + 3 \geqslant 5 $ is prime.

Prove the following:

$i)$ Prove that $f$ is irreducible and of degree $p$

$ii)$ Prove that p has exactly $p-2$ real zeros. I have no idea how to prove this, maybe with einsenstein and considering the derivate, but how? :/

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    Can you prove that $f$ has degree $p$?2012-11-18

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Modulo $2$, $f(x)$ is just $x^p$; also, the constant term of $f(x)$ is 2; thus, you can apply Eisenstein with $p=2$.

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    My derivate is bad2012-11-20