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2021-01-18 16:42:38 +0100 asked a question How can I compute a fixed field over the p-adics

I have problems to implement the following set up:

I want to have a field $K = \mathbb{Q}_3$ and an extension $L = \mathbb{Q}_3(\alpha)$ over $K$ where $f:=\min_K(\alpha) = x^4 - 3x^2 + 18$. This extension has degree $4$ and ramification index $2$. Furthermore, let $F/K$ be the unique unramified extension of $K$ of degree $4$ which is generated by a primitive $5$-th root of unity $\zeta_5$.

Then one can show that $\varphi: \alpha \mapsto \frac{(2 \alpha^2 - 3)\sqrt{-\frac{2}{7}}}{\alpha},$ $\zeta_5 \mapsto \zeta_5^3$ is an element of the Galois group of $LF/K$. Now let $L' = (LF)^{\langle \varphi \rangle}$. This must be a quadratic and totally ramified extension of $K$. There are only two possibilities for that: $K(\sqrt{3})$ or $K(\sqrt{-3})$.

Question: How to determine whether $L' = K(\sqrt{3})$ or $L' = K(\sqrt{-3})$ (or equivalently, $\varphi(\sqrt{3}) = \sqrt{3}$ or $\varphi(\sqrt{-3}) = \sqrt{-3}$)?

Since I only have only superficial knowledge about Sage, I was not even able to set up the easy things like the field $L$ properly. When I use

K = Qp(3)
R.<x> = ZZ[]
f = x^4 - 3*x^2 + 18
L.<alpha> = K.extension(f)

I get an error that my polynomial $f$ must be either unramified or Eisenstein (which of course does not exist since $L/K$ is neither unramified nor totally ramified). Furthermore, I have no idea how to approach with my problem with Sage otherwise. And since computation by hand is pretty hard in this case (I already tried!), it would be nice to solve with problem here, so I can use it for similar computations in the future.

Could you please help me with this problem?