Milne's note says when $L,K$ are number fields and $L/K$ is an abelian extension, infinitely many primes don't split. I want to ask if it's right in general case and how to prove it?
Thank you!
Milne's note says when $L,K$ are number fields and $L/K$ is an abelian extension, infinitely many primes don't split. I want to ask if it's right in general case and how to prove it?
Thank you!
Assume that all but finitely many primes split in $L/K$. Then the Dedekind zeta function $\zeta_L(s)$ agrees with $\zeta_K(s)^{(L:K)}$ apart from finitely many Euler factors. Comparing the order of the pole $s=1$, we conclude that $(L:K)=1$, that is, $L=K$.
Added. I understood split as "completely split". If split is understood as "has at least one degree one prime over it", then the above still works with some modifications. I am grateful to Levent Alpoge for this observation (see his comment below).