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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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3  
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
5  
Also, the title does not reflect the question –  Belgi Sep 22 '12 at 10:16

2 Answers 2

Hint: You can check by hand all the answers

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Hint:

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} $

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