Consider $K=\mathbb{Q}_3(\sqrt[4]{-3},i)$ and $L = K(\alpha)$ with $$ \alpha = \sqrt[3]{2} \left( \frac{(1-\zeta_3)(1-\sqrt{-7})}{6} - \zeta_3\zeta_7 + \zeta_7^2 \right) $$ where $\zeta_7$ is the primitive $7$-th root of unity with minimal polynomial $$ \min_K(\zeta_7) = x^3+\frac{1-\sqrt{-7}}{2} x^2 + \frac{-1-\sqrt{-7}}{2} x - 1.$$
Furthermore, let $v$ be the valuation on $L$ with $v(3)=1$.
Question: Is there a unit $\epsilon \in \mathcal{O}_L^\times$ satisfying the equation $\epsilon^3 \equiv \frac{1}{4}$ modulo an element of valuation $\frac{3}{2}$?
My attempts to solve the problem can be found both here and here where I managed to took the wrong $\alpha$ twice. This time the $\alpha$ should be correct (I checked!) and maybe it is helpful to know that, according to Magma, we have
- $\alpha^3 \in K$,
- $L/K$ is totally ramified of degree $3$.
Again, I am asking you to help me with this problem. Thanks in advance!