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Therem Let $f_j(x)$ be polynomials with integer coefficients.Then the following are equivalent:

(1) the polynomial system $f_j(x_1,...,x_n)=0$ for $1\leq j\leq m$ has a solution in $\mathbb{C}^n$

(2) the polynomial system $f_j(x_1,...,x_n)=0$ for $1\leq j\leq m$ has a solution modulo infinitely many primes

Is this statement true and, if that is the case, what is it called and how does one prove it? I think Hilbert's Nullstellensatz can be used in one direction and Dedekind's factorization theorem in the other, but that's just what I've heard my teacher mention (he doesn't know what the theorem is called; he just calls it a generalisation of Schur's Lemma)

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    [Chebotarev's density theorem](https://en.m.wikipedia.org/wiki/Chebotarev's_density_theorem) is probably what you are looking for. – 2017-01-28
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    See this answer [here](http://mathoverflow.net/a/118150) – 2017-01-28

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