I am working with the splitting field $L/\mathbb{Q}_2$ of the polynomial $$ h=x^4-2x^2+4\in\mathbb{Q}_2[x] $$ and want to find the inertia degree and ramification index of $L$.
Let $w$ be the unique extension of the $p$-adic valuation to $L$. Using the discriminant and resolvent cubic, I found that the Galois group of $h$ is the Klein four-group $C_2\times C_2$, and by considering the Newton polygon, it is $w(\alpha)=1/2$ for any root $\alpha$ of $h$. This means that the ramification index $e$ is at least $2$ and thus for the inertia degree it is $f\leq 4/e=2$. Additionally, the inertia group $I$ is nontrivial, because $L/K$ is ramified. There is a canonical isomorphism $G/I\to \mathrm{Gal}\left(\lambda/\mathbb{F}_2\right)$ and hence $$ |G|/|I|=\left|\mathrm{Gal}\left(\lambda/\mathbb{F}_2\right)\right|=\left[\lambda:\mathbb{F}_2\right]=f. $$
I also know that $\left(|I|/|R|,2\right)=1$ for the ramification group $R\subset I$ and thus $|I|=|R|$. Here I used that we have an injection $I/R\hookrightarrow \lambda^*$. Furthermore, because of the injections $G_k/G_{k+1}\hookrightarrow\lambda,~k\geq 1$ the factor groups $G_k/G_{k+1}$ are abelian and have order a power of $2$.
However, with all this information I still couldn't determine whether $I=C_2\times C_2$ or $I=C_2$.
In Ramification in local fields it is suggested to look at the reduction $h\equiv x^4\mod 2$, which doesn't help either (or I don't see how it would). This is the case for any polynomial whose Newton polygon has nonzero slope, which I encounter multiple times.
Is there a way to determine the inertia group $I$ with the information I provided, or is there a different/better way? I mostly worked with Chapter 2 of Neukirch's Algebraic Number Theory.
In the local fields database https://math.la.asu.edu/~jj/localfields/ we can see that for the polynomial $h$ it is $e=f=2$.
Also, for the polynomial $$ h_2=x^4+3x^2+1\in\mathbb{Q}_2[x] $$ we have $$ h_2\equiv x^4+x^2+1 \equiv \left(x^2+x+1\right)^2 \mod 2. $$ From how I understood the comments in Ramification in local fields, this already implies $e=f=2$ for the stem field of $h_2$. Is this correct? If it is correct, could you please suggest a source where I can find the involved results?
Thank you very much for your help!