Let $L_k = \mathbb{Q}(\zeta_{2^k} + \zeta_{2^k}^{-1})$ be the maximal real subfield of the cyclotomic field of conductor $2^k, k \ge 2$ and $f_k(x)$ be the minimal polynomial of $\zeta_{2^k} + \zeta_{2^k}^{-1}$.

Define $L = L_{k+1}, K = L_{k}$ so $L/K$ has degree 2. Assume a prime ideal $\mathfrak{p} \subset \mathcal{O}_K$ above $p$ splits in $\mathcal{O}_L$ as $\mathfrak{p}\mathcal{O}_L = \mathfrak{p}_1 \mathfrak{p}_2$. Then I suspect that $p\mathcal{O}_L$ totally splits (in which case $p\mathcal{O}_K$ totally splits as well). How can I prove this?

I believe this is equivalent to showing that if $f_{k+1}(x)$ has 2 roots over $\mathbb{F}_p$ then it factors completely over $\mathbb{F}_p$. I'm not sure how that can be shown. Are there any general approaches to take here?