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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 $\mathbf{Q}(\zeta_N)$. \mathbf{Q}(\zeta_N)$). But if$1\not\in S$then no only a finite number of primes at all 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. 1 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.