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2 biquadratic residue mod p

  1. Feb 17, 2005 #1
    show that x^4 == 2( mod p) has a solution for p==1(mod 4) iff
    p is of the form A^2+64B^2, where A,B are integers


    I let x^2=M
    then the conguence is reduced to M^2==2( mod p)
    but any # squared == 0 or 1 ( mod4 ) so p must be == 1(mod4)...
    but i'm not sure what to do now..
    any hints/ or ideas ?
     
  2. jcsd
  3. Feb 18, 2005 #2

    matt grime

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    Find when 2 is a residue mod p using, say, Quadratic Reciprocity, and then find when its square root is also residue, again using reciprocity
     
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