Let $K = \mathbb{Q}_3$ and $\zeta_8$ a primitive $8$-th root of unity.
Question: What is $\min_K(\zeta_8)$?
I know that the extension $K(\zeta_8)/K$ is unramified of degree $2$, so the degree of the minimal polynomial must be $2$.
I also know that the minimal polynomial must be a divisor of $x^4+1$, the $8$-th cyclotomic polynomial over $\mathbb{Q}$. This polynomial also factorizes in two quadratic polynomials in $\mathbb{F}_3$ which I checked with Wolfram Alpha.
Could you please help me advancing with this problem?