$ \phi : f(x, y) \mapsto f(x, x^2) $ from the polynomial ring $ \mathbb{C}[x, y] $ to the polynomial ring $ \mathbb{C}[x] $. Determine its kernel.

Asked Jan 02 '25 at 12:42

Active Jan 02 '25 at 12:42

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Consider the map $ \phi : f(x, y) \mapsto f(x, x^2) $ from the polynomial ring $ \mathbb{C}[x, y] $ to the polynomial ring $ \mathbb{C}[x] $. Show that $ \phi $ is a homomorphism of rings and determine its kernel.

Showing that $ \phi $ is a homomorphism is trivial. It is also easy to see that $(y-x^2)$ $\subseteq$ ker($\phi$), where $(y-x^2)$ denotes the principal ideal generated by $y-x^2$. However, I don't know what other elements the kernel might have or whether ker($\phi$) $=$ $(y-x^2)$.

Thanks in advance.

asked Jan 02 '25 at 12:42

Mustafa

2

This is precisely the kernel. Hint: If we denote $I=(y-x^2)$, then in the quotient $\mathbb{C}[x,y]/(y-x^2)$ we have $y+I=x^2+I$, and hence $f(x,y)+I=f(x,x^2)+I$. – Mark Jan 02 '25 at 12:46
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@AnneBauval This question will probably be closed as a duplicate anyway. Even I already answered questions of this type, maybe I can find it. – Mark Jan 02 '25 at 12:53
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This question is similar to: the kernel of the evaluation map. Apply the latter to $R:=\Bbb C[x]$ and $e_{x^2}:R[y]\to R$. – Anne Bauval Jan 02 '25 at 13:02

$ \phi : f(x, y) \mapsto f(x, x^2) $ from the polynomial ring $ \mathbb{C}[x, y] $ to the polynomial ring $ \mathbb{C}[x] $. Determine its kernel.

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