Dear all, I would be grateful if someone could provide a solution to the following problem (using decomposition and inertia groups):
Find a finite extension of $\mathbb{Q}$ in which all primes split.
[Hint: Use the fact that a prime splits if and only if its decomposition group is not the full Galois group (and that the decomposition group is cyclic for all unramified primes)]
Many thanks, Mohammad.