Question

Determine the Hilbert symbol \[\left( \frac{2,0}{\mathbb F_{25}} \right)\] where 'F' denotes the field with 5² elements

Answer 1

\[\left( \frac{2,0}{\mathbb F_{5}} \right) = -1\]

so f(x) = 2x²-1 has no solution in F_5, which means you can't reduce it. Let's say the splitting field of the polynomial is 'X.' So by construction:

\[\left( \frac{2,0}{X} \right) = 1\]

X is a vector space over F_5 of 2nd dimension and the degree of f is 2 so it has to be isomorphic to F_25.

Am I missing anything?

Answer 2

Actually I'm unsure of the latter statement