I've been puzzled about some basic facts in (classical) algebraic geometry, but I cannot seem to find the answer immediately:
Let $V=V(f_1,\ldots,f_n)$ be a variety over some field $k$, and let $n > 1$. Suppose that $V$ turned out to be reducible, i.e. it is the union of irreducible varieties $V_1,\ldots,V_m$ for some $m > 1$. Must it be the case that at least one of the $f_i$s are reducible polynomials?
Take one of the irreducible components, $V_1$, say. Do the defining equations for $V_1$ have anything to do with the polynomials $f_1,\ldots,f_n$?
Suppose varieties $V = V(f_1,f_2)$ and $W = V(f_3)$ shared a common component. Does that mean that $f_3$ shares a factor with either one of $f_1$ or $f_2$? If not, what's a counterexample?
Thanks so much!