This equivalent to $f(x)=x^{4}-2x^2 +3$ being Eiseinstein. We have $|1|_{2}=1$, $|2|_{2}=2^{-1}$ and $|3|=|1|$. So 3 is a 2-adic unit. Thus, f is not Eisenstein and so L is not completely ramified. Where is my mistake?
2) Next is showing that $1,\sqrt{-2},\sqrt{1+\sqrt{-2}},\sqrt{-2}\sqrt{1+\sqrt{-2}}$ is a basis for ring of integers $O_{L}$. Any suggestions?
Thanks