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This is an exercise in Artin's Algebra (10.3.11)

$\phi :\mathbb{C}[x,y] \rightarrow \mathbb{C}[t]$ is defined by $\phi(x)=t^2$ and $\phi(y)=t^3$

Prove the $\operatorname{Ker} \phi =$ the principal ideal generated by $y^2-x^3$

and describe the $\operatorname{Im} \phi$ explicitly

I am able to show the Ideal generated by y^2-x^3$ is contained in the kernel but don't know how to go about the other containment or show it is a principal ideal.

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    The ideal generated by such-and-such element is principal by definition, so the only thing left to show is that the kernel is in the ideal.2011-10-13

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Hint: Let $I$ be the ideal generated by $y^2-x^3$. Show that every coset $g+I$ contains a unique element of the form $f_1(x)+yf_2(x)$ with $f_1,f_2\in \mathbf{C}[x]$. Then show that such a polynomial is in $\ker \phi$ only, if $f_1=f_2=0$.