I have few questions which are related to a problem I am solving. Let $v = (v_1,\dots, v_n)$ be a fixed vector in $\mathbb{C}^n$ and $n>1$.
[1] Is $\mathbb{R}$ an algebraic set in $\mathbb{C}^n$ for $n>1$? Prove/disprove (by giving a counterexample).
[2] Is $\mathbb{C}$ the smallest uncountable (or probably even countably infinite) algebraic subset in $\mathbb{C}^n$?
[3] Claim: Let $f: \mathbb{C}^n \rightarrow \mathbb{C}$ s.t. $f=0$ for all the points $\{mv: m \in \mathbb{Z}^+ \}$. Is it true that $f=0$ for all $\{mv: m \in \mathbb{C} \}$. Is there any theorem in complex analysis that has such a result.