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I understand that uni-variate polynomial rings with coefficients in a field only have principal ideals. For example, $\mathbb{C}[x]$. But how can I tell if an ideal of integer polynomial ring is principal, please? For example, a textbook claims that "the kernel of the map $\mathbb{Z[x]} \rightarrow \mathbb{Z[i]}$ sending $x \mapsto i$ is the principal ideal of $\mathbb{Z}[x]$ generated by $f=x^2+1$" without any justification. How to show this is true, please?

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The quote tells you the element $f\in Z[x]$ that generates the ideal. Suppose now another $g\in Z[x]$ is in that ideal. Since $f$ is monic you can use Eucledian algorithm to find $h,r\in Z[x]$ such that $g=fh+r$, and $deg(r)\leq 1$. Since degree 0 or 1 polynomial are in said kernel conclude that $r\equiv 0$, therefore all elements $g$ of the ideal have form $fh$ for some $h$, therefore the ideal is generated by $f$.

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