As the title says, I'm trying to find ideals of $\mathbb{Z}[x]$ generated by $n$ elements and no fewer. I suspect $(2^k, 2^{k-1} x, 2^{k-2} x^2, ..., x^k)$ is generated by no fewer than $n=k+1$ elements, but I haven't been able to prove it. I've tried successively replacing single elements in a finite generating set in an attempt to zero out the constant order coefficients of all but one generator, but I'm not able to ensure I've done so while preserving the ideal. If I could do this, an inductive proof would follow immediately. I've also tried to make other examples but none seemed as promising as this one. This one also generalizes the neat example $(2, x)$ (generated by at least 2 elements) which shows explicitly that $\mathbb{Z}[x]$ is not a PID. I've searched and found nothing useful.

This is problem 3.7 from D.J.H. Garling's *A Course in Galois Theory*. The chapter itself seems to be a standard introduction to commutative algebra with an eye towards polynomial rings over fields. I hope the question isn't too basic for this site; it's my first post. I've gone through the book and done every problem except for (parts of) around a dozen, including this one. He gives questions that appear to rely on more advanced material than was presented in the text from time to time, so perhaps this question is easy for someone more experienced in algebra, or maybe I'm just missing something.

Any help is appreciated!