Why is $I = \left \{ \sum_{i=0}^n a_i x^i | a_0 \in 2 \mathbb{Z} \right \} \subseteq \mathbb{Z}[x]$ not a principal ideal?

Question

Why is $I = \left \{ \sum_{i=0}^n a_i x^i | a_0 \in 2 \mathbb{Z} \right \} \subseteq \mathbb{Z}[x]$ not a principal ideal?

We saw it as a short example for a non-principal ideal in a linear algebra course, I have no knowledge in abstract algebra.

But I think every polynom can be written as $2 *p(x)$ for some $p$ so I'd expect it to be principal, what am I missing?

I barely understand the language of this field so a basic answer would be best.

Answer 1

You are missing that $x\in I$ as well as $2\in I$. There is no single generator that gives both without also giving $1\in I$.