Let $f(x_1,\ldots,x_n)\in\mathbf{Z}[x_1,\ldots,x_n]$ be a polynomial. Assume that the variety cut out by $f$ is smooth and connected (so irreducible) over $\overline{\mathbf{Q}}$. Where can I find a proof of the following classical result: for almost all primes $p$, $f\pmod{p}$ is a smooth $\mathbb{F}_p$-scheme.

Let $f(x_1,\ldots,x_n)\in\mathbf{Z}[x_1,\ldots,x_n]$ be a polynomial. Assume that the variety cut out by $f$ is smooth and connected (so irreducible) over $\overline{\mathbf{Q}}$. Where can I find a proof of the following classical result: for almost all primes $p$, $f\pmod{p}$ is a smooth $\mathbb{F}_p$-scheme.

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Let me rephrase the problem a bit differently. Let us denote the zero locus in a field $k$ of a family of polynomials $\mathcal{F}$ with $\mathbf{Z}$-coefficients by $Z(\mathcal{F};k)$. Let $\nabla(f)=(g_1,g_2,\ldots,g_n)$. So assume that $$ Z(g_1,g_2,\ldots,g_n,f;\overline{\mathbf{Q}})=\emptyset $$ How do you prove that for almost all primes $p$ $$ Z(g_1,g_2,\ldots,g_n,f;\overline{\mathbf{F}_p})=\emptyset ? $$

Let $f(x_1,\ldots,x_n)\in\mathbf{Z}[x_1,\ldots,x_n]$ be a polynomial. Assume that $f$ is a smooth connected (so irreducible) over $\overline{\mathbf{Q}}$. Where can I find a proof of the following classical result: for almost all primes $p$, $f\pmod{p}$ is a smooth $\mathbb{F}_p$-scheme.

Let $f(x_1,\ldots,x_n)\in\mathbf{Z}[x_1,\ldots,x_n]$ be a polynomial. Assume that the variety cut out by $f$ is smooth and connected (so irreducible) over $\overline{\mathbf{Q}}$. Where can I find a proof of the following classical result: for almost all primes $p$, $f\pmod{p}$ is a smooth $\mathbb{F}_p$-scheme.

Let $f(x_1,\ldots,x_n)\in\mathbf{Z}[x_1,\ldots,x_n]$ be a polynomial. Assume that $f$ is irreducible over $\overline{\mathbf{Q}}$. Where can I find a proof of the following classical result: for almost all primes $p$, $f\pmod{p}$ is a smooth $\mathbb{F}_p$-scheme.

Let $f(x_1,\ldots,x_n)\in\mathbf{Z}[x_1,\ldots,x_n]$ be a polynomial. Assume that $f$ is a smooth connected (so irreducible) over $\overline{\mathbf{Q}}$. Where can I find a proof of the following classical result: for almost all primes $p$, $f\pmod{p}$ is a smooth $\mathbb{F}_p$-scheme.