<s>[INST] Multiple roots of a polynomial over a field of characteristic $p$ I have to show for what value of the prime $p$ does the polynomial $ x ^4 + x + 6$
 have a root of multiplicity $>1$ over the field of characteristic $p$.
$ p=2, 3, 5,  7 $ 
Please help. 
For $F$ a field of characteristic $3$, $f(x)= x^4 + x = x(x^3+1)$ and $f'(x) = x^3+1$. Hence, $f^′(x)= 0$ for $x=2$. Therefore in an algebraically closed field of characteristic $3$, $f(x)$ has multiple roots. 
 [/INST] The discriminant of $f(x)$ is $55269 = (3^3)(23)(89)$.  So the characteristics in which $f(x)$ has a multiple root are $3$, $23$ and $89$.  In fact, $f(x) = x(x+1)^3$ in characteristic $3$, $(x^2+7x+8)(x+8)^2$ in characteristic $23$ and $(x+8)^2(x+28)(x+45)$ in characteristic $89$.
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