I'm currently in the middle of teaching the adelic algebraic proofs of global class field theory. One of the intermediate lemmas that one shows is the following:

**Lemma:** if L/K is an abelian extension of number fields, then there are infinitely many primes of K that do not split competely in L.

Of course this is implied by Cebotarev's density theorem, but the adelic proof uses only algebra/topology and finiteness of class number/Dirichlet's units theorem.

There is a well-known elementary proof, (see eg this MO question) that there are infinitely many primes that *are* split in L/K. I was wondering whether there is also an elementary argument for infinitude of non-split primes in the extension? (As usual the notion of "elementary" is flexible, but I'm looking for something that uses a minimum of machinery.)

One possibility would be to distill the adelic proof into something algebraic, although this seems hard. Another option would be to look for ideals of O_K that are not norms from O_L: any such ideal must have a factor which does not split completely.

One of the answers to the MathOverflow question linked above does mention the paper *Primes of degree one and algebraic cases of Čebotarev's theorem* of Lenstra and Stevenhagen, which gives an elementary proof under the assumption that L contains a nontrivial ray class field of K. But it seems that one still needs to prove the first inequality in some form to use this.