Can someone help with the following exercise, I really don't know how to approach this:
Let $f_1,...,f_k \in \mathbb{Z}[x_1,...,x_n]$ be polynomials without common zero $(a_1,...,a_n) \in \mathbb{C}^n$.
Show that there exist $g_1,...,g_k \in \mathbb{Z}[x_1,...,x_n]$ with $0 \neq g_1f_1+...+g_kf_k \in \mathbb{Z}$
Is this still true if $\mathbb{C}^n$ is replaced by $\mathbb{R}^n$?