- #1
nonequilibrium
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- 2
Homework Statement
Determine the Hilbert symbol [itex]\left( \frac{2,0}{\mathbb F_{25}} \right)[/itex] where the F denotes the field with 5² elements.
Homework Equations
[itex]\left( \frac{2,0}{\mathbb F_{5}} \right) = -1[/itex]
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?)