Let $f(t) \in \mathbb{C}[t]$.
In the polynomial ring $\mathbb{C}[x,y]$, denote by $I_{f(x),y}$ the ideal generated by $f(x)$ and $y$, $I_{f(x),y}:= \langle f(x),y \rangle$.
Recall that an ideal $I$ in a ring $R$ is called radical, if $a^n \in I$ implies that $a \in I$.
When $\langle f(x),y \rangle$ is a radical ideal? (= for which $f(t) \in \mathbb{C}[t]$).
Examples:
(1) Take $f(t)=t^2$. Then $I_{x^2,y}=\langle x^2,y \rangle$ is not a radical ideal, since $x^2 \in I_{x^2,y}$ but $x \notin I_{x^2,y}$. Similarly, for $f(t)=t^m$, $I_{x^m,y}$ is not a radical ideal.
(2) For $f(t)=t$, the ideal $I_{x,y}$ is radical: $I_{x,y}$ is a maximal ideal (by Hilbert's Nullstellensatz), so it is a prime ideal (every maximal ideal is prime) and clearly a prime ideal is a radical ideal (follows immediately from the definitions).
Any hints and comments are welcome!