Sorry if the question is too long and maybe elementary.
I am reading a paper by Hirotada Naito on "Dihedral extensions of degree 8 over the rational p-adic fields". To generate dihedral extension $K(\sqrt{\epsilon},\sqrt{\epsilon^\sigma})$ in part 2-1, he said let $K=\mathbb{Q}_2(\sqrt{m})$ for $m=2,-2, 10$ or $-10$ $$K^*/K^{*2}\cong (\langle\sqrt{m}\rangle/\langle m\rangle)\times (\mathcal{O}^\times/\langle 1+m+2\sqrt{m}, 1+ \mathfrak{p}^5 \rangle)$$ for $\mathfrak{p}= \langle\sqrt{m}\rangle$.
And after that, he said "it is sufficient to examine $\epsilon$ and $\epsilon\sqrt{m}$ where $\epsilon=a+b\sqrt{m}$ for $a=1,3,5,7$ and $b=0,1,2,3$."
My first question is "why it is enough to check these numbers?"
In the following, he said "we take $\epsilon$ (resp. $\epsilon\sqrt{m}$) such that $\epsilon$, $\epsilon^\sigma$, $\epsilon (1+m+2\sqrt{m})$ and $\epsilon^\sigma (1+m+2\sqrt{m})$ (resp. $\epsilon$, $-\epsilon^\sigma$, $\epsilon (1+m+2\sqrt{m})$ and $-\epsilon^\sigma (1+m+2\sqrt{m})$) are different modulo $\mathfrak{p}^5$ each other." Where $\sigma$ is the generator of Galois group of $K/\mathbb{Q}_2$.
And in the following, for $m=2$ he took $1+\sqrt{2}$, $3+\sqrt{2}$, $\sqrt{2}$ and $3\sqrt{2}$. I couldn't get to these numbers following his method explained above, can anyone show me the calculations? And can you show me any other possible numbers other than the given ones?
Here is the paper: Dihedral extensions of degree 8 over the rational p-adic fields. Thank you.