Let $k$ be an algebraically closed field and $V_1$, $V_2$ are two irreducible affine varieties. Suppose we have finite maps $f_i: V_i\to \mathbb{A}^1(k)$. How can we prove that the fiber product $V_1\times_{\mathbb{A}^1(k)} V_2$ is again irreducible? If not then what conditions will enable the fiber product irreducible.
Here finite map means the map on the coordinate rings are finite.
Thank you in advance