<s>[INST] Suppose that poynomial $x^4+x+1$ has multiple roots over a field of characteristic $p$ . What are the possible values of $p$ 
Suppose that polynomial $x^4+x+1$ has multiple roots over a field of
  characteristic $p$  . What are the possible values of $p$?

My solution : 
Set $f=x^4+x+1$. Suppose the multiple root is $m$ . So $f,f'$ (the formal derivative of $f$) have root $m$ in the field of characteristic $p$. 
Hence $m^4+m+1=0 \pmod{p} $ and $4m^3+1=0 \pmod{p}$ 
So $3m+4=4(m^4+m+1)-m(4m^3+1)=0 \pmod{p}$ .
From here it is easy to to see that $p\neq 3$. I have no idea if it is possible to proceed any further. Please provide a solution. 
 [/INST] Using the extended Euclidean algorithm we get
$$
229 = (x^4 + x + 1)(144 x^2 - 192 x + 256)+(4 x^3 + 1)(-36 x^3 + 48 x^2 - 64 x - 27)
$$
This can also be found by computing the resultant of $x^4 + x + 1$ and $4 x^3 + 1$, that is, the discriminant of $x^4 + x + 1$.
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