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It is fairly known that

$\mathbb C[x,y,z]/(xy+z^n) \cong \mathbb C[x,y,z]/(z^n+x^2+y^2)$.

This appears, for example, in the study of singularities of type $A_n$. But, unfortunately, I am not able to prove it.

How could we establish this isomorphism?

Thanks!

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You should edit the title so that it refers to the content of the question, which is not about isomorphisms between polynomial rings! :-) –  Mariano Suárez-Alvarez Oct 29 '12 at 21:12

2 Answers 2

up vote 4 down vote accepted

Hint: The equality $x^2+y^2=(x+iy)(x-iy)$ suggests $x\mapsto x+iy$, $y\mapsto x-iy$, $z\mapsto z$.

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Thank you! That was easier than I thought! –  Ferenc Oct 29 '12 at 21:11

In the left hand side, set $x=u-iv$, $y=u+iv$. Then the LHS is $$\mathbb C[u,v, z]/((u^2+v^2)+z^n).$$

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Thank you! This is useful! –  Ferenc Oct 29 '12 at 21:11

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