Generator for $\mathbb{Z}[x]$ ring?

Question

Is the polynomial ring $\mathbb{Z}[x]$ generated exclusively by $\langle 1,x\rangle$? Or has to be generated by more or less elements? My intuition is that you can generate any of the "$x$'s" polynomials with just multiplying and adding $x$ to the equation but you need the "$1$" to generate the coefficient with degree $0$.

Answer 1

Your intuition would be correct. The only way to generate $\mathbb{Z}\subset \mathbb{Z}[x]$ is by having 1 in your ideal. Similarly, you must have x in your ideal to generate all the polynomials.