I have a question about square classes of quadratic extensions of 2-adic fields. I appreciate anybody help me to understand.
Why all elements of $1+\mathfrak{p}^5$ are square in $\mathbb{Q}_2(\sqrt{m})$ where $\mathfrak{p}=\langle \sqrt{m} \rangle$ for $m=\pm 2 \pm 10$ and $\mathfrak{p}=\langle 1+\sqrt{m} \rangle$ for $m=-1, -5$?