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Find a degree 5 polynomial $f \in \mathbb Z_5[x]$ so that it has exactly 4 distinct roots and factorize it as product of irreducible factors.

I'm really struggling in finding such polynomial, so basically I need to find an $f$ which has a $c$ root that doesn't belong to $\mathbb Z_5$. So does this polynomial is the one I am looking for? If not is there any way I could get to such $f$?

$f = x^5+\sqrt{2}x^4-5x^3-5\sqrt{2}x^2+4x+4\sqrt{2} = (x+1)(x-1)(x+2)(x-2)(x+\sqrt{2})$

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    Your example is *not* a pol. in $\,\Bbb Z_5[x]\,$ since $\,\sqrt 2\notin\Bbb Z_5\,$2012-07-15
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    Is it clear that multiple roots are disallowed? Can't you say that, e.g., $x^3-x^2$ has two distinct roots (and three when counting with multiplicity)?2012-07-15
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    In English, not "grade" but "degree".2012-07-15
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    @PerManne I'm sorry, but I'm afraid I don't understand the point you're making.2012-07-15
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    @GEdgar I was in a rush and I didn't pay much attention to the language, I apologize.2012-07-15
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    @haunted85 The example $x^3-x^2$ was supposed to be a little simpler than your case. The factorization is $x^3-x^2=x\cdot x\cdot(x-1)$, so it is a third degree polynomial with two distinct roots given by $x=0$ and $x=1$. A similar expression will give you a fifth degree polynomial with four distinct roots.2012-07-15

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