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[Number theory] Calculate the Hilbert symbol

  1. Jun 9, 2012 #1
    1. The problem statement, all variables and given/known data
    Determine the Hilbert symbol [itex]\left( \frac{2,0}{\mathbb F_{25}} \right)[/itex] where the F denotes the field with 5² elements.

    2. Relevant equations
    [itex]\left( \frac{2,0}{\mathbb F_{5}} \right) = -1[/itex]

    3. The attempt at a solution
    Due to the formula that I put under "relevant equations", we know that the polynomial f(x) = 2x²-1 has NO solution in F_5, hence it is irreducible. Look at the splitting field of this polynomial, call it X. Then by construction [itex]\left( \frac{2,0}{X} \right) = 1[/itex]. Now note that since the degree of f is 2, X is a vector space over F_5 of dimension 2 and hence has to be isomorphic to F_25. This concludes the proof.

    (The last part is what I'm unsure about; does it require more argumentation? And is there perhaps an even shorter way of showing that the Hilbert symbol is 1?)
     
  2. jcsd
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