Is there a non-constant $h \in \mathbb{C}[x_1 , \dots , x_n ]$ that divides every element of this given ideal?

Question

I'm trying to prove the following:

Let $I ⊆ \mathbb{C}[x_1 , \dots , x_n ]$ be an ideal with the property that any two elements $f_1$ , $f_2 \in I$ have a non-trivial common divisor. Then there is a non-constant $h$ which divides every element of $I$.

I tried to use the fact that $\mathbb{C}[x_1 , \dots , x_n ]$ is a Noetherian ring to do induction on the number of generators of $I$. But I did not succeed.

Can anyone help me?