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?



How can I prove this claim about splitting of prime ideals in real cyclotomic fields?
Use that $\text{Gal}(L\_k/\mathbb{Q})$ is cyclic and look at the fixed field of the decomposition group.


