[Number theory] Calculate the Hilbert symbol

1. Jun 9, 2012

nonequilibrium

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

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

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 $\left( \frac{2,0}{X} \right) = 1$. 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?)