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Math Help - Determine solvablity

  1. #1
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    Determine solvablity

    Prove that x^{2}+2\equiv0 \mod p is unsolvable if p\equiv 5,7\mod 8 .
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  2. #2
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    Quote Originally Posted by chipai View Post
    Prove that x^{2}+2\equiv0 \mod p is unsolvable if p\equiv 5,7\mod 8 .
    Thanks
    Can we use Gauss's Lemma ?

    To determine whether the given number a \neq 0  is quadratic residue , consider the list :

      a , 2a , 3a , ..., Pa  ~,~ P = \frac{p-1}{2}

    If the number of the elements in the above list lying between  P+1 and  2P inclusively is even , then  a is quadratic residue , otherwise , it is quadratic non-residue .

    In this case ,  a \equiv  -2 \bmod{p} and  P is either  2\bmod{4} or  3 \bmod{4}

    The elements lying between the range should be :

     -2 , -4 , .... , (-2) \frac{P}{2} for  P \equiv 2 \bmod{4} so the number is  \frac{P}{2} an odd number .

    Or

     -2 , -4 , ..., (-2) \frac{P-1}{2} for  P \equiv 3 \bmod{4} so the number is  \frac{P-1}{2} also an odd .

    Therefore ,  - 2 is not the quadratic residue of modulo  p \equiv 5,7 \bmod{8} .

    Remarks : so we can see that  - 2 is quadratic residue of modulo  p \equiv 1,3 \bmod{8} .
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