Let $K=\mathbb{Q}_2$ and $f = x^4 - 2x^2 - 1/3 \in K[x]$.
Suppose that $L$ is the splitting field of $f$ over $K$. With Magma, I computed that the residue field of $L$ is given by $\mathbb{F}_4 = \mathbb{F}_2(\alpha)$, where $\alpha$ is an element with minimal polynomial $x^2 + x + 1$ over $\mathbb{F}_2$.
Further, I noticed that the reductions of all four roots of $f$ in $L$ are equal to $1$.
Question: How can I find an element $\beta \in L$ such that the reduction of $\beta$ is $\alpha$, i.e. $\bar{\beta} = \alpha$?
Since the reductions of all four roots of $f$ in $L$ are equal to $1$, I probably need to combine these roots to obtain such a $\beta$ - now I am interested in how to do this in a systematic way. It would be nice if you have any advice for this kind of problem.