# Characteristic of Field

For which prime $p$, the polynomial $x^4 + x + 6$ have a root of multiplicity $> 1$ over a field of characteristic $p$?

Options:

1. 2
2. 3
3. 5
4. 7
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Is this homework ? what did you try ? what do you know about when a polynomial have multiple roots ? –  Belgi Sep 22 '12 at 10:10
Also, the title does not reflect the question –  Belgi Sep 22 '12 at 10:16

Hint: You can check by hand all the answers

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If characteristic is 2 or 3 then the equation will be $x(x^3 + 1 )$ So consider $x^3+1$ in case of $char \space 2$ one is a root of $x^3+1$, so factor $x^3+1$ and check if 1 is root of factor, if not check for zero.
If above thing fails, so the same with $Char \space3$, here 1 is not a root of $x^3+1$, so only options are 2 and 0. In fact 2 is a root of $x^3+1 = 8+1 = 0$ so factor $x^3+1$ with $\pmod 3$ and check.
For 5 and 7 you can minimize calculation overhead by considering ${ -2, -1, 0, 1 ,2}$ and ${ -3,-2, -1, 0, 1 ,2, 3}$