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So I currently have this question:

Does there exists $\alpha \in \mathbb{C}$ such that the kernel of $e.v._{\alpha}: \mathbb{C}[x] \to \mathbb{C}$ is $\{(x^2+x+1)f(x) \vert f(x) \in \mathbb{C}[x] \}$ ?

The question before this was similiar but instead it was $e.v._{\alpha}: \mathbb{R}[x] \to \mathbb{C}$ for which I found such $\alpha$'s. I don't why there is a difference between these two. Am I missing something or just overthinking?

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    What about the roots of $x^2+x+1$? – 2017-02-14
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    That's what I figured worked in the first case in $\mathbb{R}[x] \to \mathbb{C}$ and thus I assume the same roots would work for the second case. If that is true, then there really isn't a difference in the questions. @PeterMelech – 2017-02-14
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    Well, the roots are complex values. @AndreasCaranti – 2017-02-14
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    Actually, no, the question said, "find all $\alpha \in \mathbb{C}$ such that the kernel of $e.v_{\alpha} : \mathbb{R}[x] \to \mathbb{C}$ is..." . @AndreasCaranti – 2017-02-14
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    @ElSpiffy I´m sure the same roots work here – 2017-02-14
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    Alright, then I was just overthinking. Thank you. @PeterMelech – 2017-02-14
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    No wait, there doesn´t exist such $\alpha\in\mathbb{C}$, the kernel would be the ideal generated by $x-\alpha$ for each root – 2017-02-14
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    @PeterMelech I have barely touched the ideal topic in my course so I am not quite sure what you mean. Would you mind elaborating? – 2017-02-14
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    Let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/53603/discussion-between-peter-melech-and-el-spiffy). – 2017-02-14

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The only possible numbers would be the roots of $x^2+x+1$, say $\alpha$ and $\overline{\alpha}$. The kernel of $e.v._{\alpha}$ is $\{(x-\alpha)f(x):f\in\mathbb{C}[x]\}$ and the kernel of $e.v._{\overline{\alpha}}$ is $\{(x-\overline{\alpha})f(x):f\in\mathbb{C}[x]\}$ both of which are obviously different from $\{(x^2+x+1)f(x):f\in\mathbb{C}[x]\}$. The point is that $x^2+x+1$ is irreducible over $\mathbb{R}[x]$ but of course not over $\mathbb{C}[x]$ because $\mathbb{C}$ is algebraically closed. So there doesn´t exist such a number $\alpha\in\mathbb{C}$