How does an ideal generated by $(x^{2})$ in $Z[x]$?

Question

I was just wondering whether there would exist an element $x^{-2}$ and how would the elements in general look like?

Answer 1

It's all the polynomials in $\Bbb Z[x]$ with zero constant and $x$-term.

$x^{-2}$ is not a polynomial, so not an element of $\Bbb Z[x]$.

Answer 2

The ideal generated by $x^2$ in $\Bbb Z[x]$ is the set of all polynomials which may be written as $x^2f(x)$ for some polynomial $f\in\Bbb Z[x]$. Specifically, if $g(x)=\sum_{j=0}^m a_jx^j$, then $g\in(x^2)$ if and only if $a_0=a_1=0$.