I know that discriminant tells exactly which prime ramify in an extension, and it helps to construct extensions where
a certain set of primes will ramify.
But I don't know how to construct extensions where certain set of primes splits completely, more generally I am trying to solve the following problem -:
Let $K$ be a number field and let $P_1, P_2, P_3,…,P_m$ be some set of primes in $O_K$. For any given $n$, Is it possible to construct an extension $L$ of $K$ of degree $n$ such that these primes splits completely in $O_L$?.
Best Answer
Take $m$ such that $n\le \inf_j N(P_j)^m$
Using the CRT, take $f\in O_K[x]$ monic of degree $n$ such that $f\bmod Q$ is irreducible and such that $f\equiv \prod_{i=1}^n (x-a_{ij})\bmod P_j^{2 (nm)^2}$ where the $a_{ij}$ are distinct $\bmod P_j^m$.
This will ensure that $f$ is irreducible in $K[x]$ and (with the extended Hensel lemma) that it splits completely in the $p$-adic completions $K_{P_j}$.
Then let $L = K[x]/(f)$.