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Let $k$ be a field. I want to show that $k[x,y]\ncong k[u,v,w]/(uw-v^2)$ as $k$-algebras, but can't find a way to do it.

The dimension of the $k$-vector space generated by the degree 1 monomials are different on both sides, but then it's possible that an isomorphism doesn't preserve the graded parts, right?

Any help will be appreciated.

user26857
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Hajime_Saito
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1 Answers1

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Hint; $k[x,y]$ is a unique factorization domain but $k[u,v,w]/(uw-v^2)$ is not a UFD, since $[v]^2=[u][w]=[v][v].$

Tsemo Aristide
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