OK how's about this to finish (I don't think either argument posted so far deals with this case). Say $K/\mathbf{Q}$ is finite and (away from a finite set of exceptions) $p$ splits completely in $K$ iff $p$ mod $N$ is contained in a subset $S$ of $(\mathbf{Z}/N\mathbf{Z})^\times$. I think the other two answers just deal with the case when $1\in S$ (where they show $K$ is contained in $\mathbf{Q}(\zeta_N)$. But if $1\not\in S$ then no primes at all split in the compositum of the Galois closure of $K$ and $\mathbf{Q}(\zeta_N)$ and that's a contradiction. So now I think between us we have completely answered the question.

OK how's about this to finish (I don't think either argument posted so far deals with this case). Say $K/\mathbf{Q}$ is finite and (away from a finite set of exceptions) $p$ splits completely in $K$ iff $p$ mod $N$ is contained in a subset $S$ of $(\mathbf{Z}/N\mathbf{Z})^\times$. I think the other two answers just deal with the case when $1\in S$ (where they show $K$ is contained in $\mathbf{Q}(\zeta_N)$). But if $1\not\in S$ then only a finite number of primes split completely in the compositum of the Galois closure of $K$ and $\mathbf{Q}(\zeta_N)$ and that's a contradiction. So now I think between us we have completely answered the question.