This question comes from Borevich-Shafarevich Number Theory CH.1 Sec.6, from which I am teaching myself about p-adic numbers.
It asks you to find all 5-adic numbers represented by the form $f = 2x^2 + 5y^2$
I have used the Hilbert Symbol to find that $(5,2)_5 = -1$ , and therefore $2x^2 + 5y^2 -z^2$ does not represent $0$, ie I know that the form f does not represent squares in $\mathbb{Q}_5^*$. I have also considered finding an equivalent form and looking at its representations but am not sure this would be much help.
Thank you in advance.
If $\infty\ne n=v(x)\le v(y)$ then $2x^2+5y^2 \in 5^{2n} 2 (\pm 1+5\Bbb{Z}_5)$. Conversely any element of $5^{2n} 2 (\pm 1+5\Bbb{Z}_5)$ is represented by $2x^2$.
If $\infty\ne m=v(y)< v(x)$ then $2x^2+5y^2 \in 5^{2m+1} (\pm 1+5\Bbb{Z}_5)$. Conversely any element of $5^{2m+1} (\pm 1+5\Bbb{Z}_5)$ is represented by $5y^2$.
So the quadratic form represents $$0\cup 5^{2\Bbb{Z}} (\pm 2+5\Bbb{Z}_5)\cup 5^{2\Bbb{Z}+1} (\pm 1+5\Bbb{Z}_5)$$