How can I compute $\sqrt{(x^2+y^2-1,yz-1)}$ as ideal of $\mathbb{C}[x,y,z]$?
Actually I have to prove that $(x^2+y^2-1,yz-1)$ is prime but I don't know how. Could you give me some suggestions, please?
2026-04-04 12:09:16.1775304556
Radical of an ideal.
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We have isomorphisms of rings $$ \mathbb C[x,y,z]/(x^2+y^2-1,yz-1) \simeq \left( \mathbb C[x,y]/(x^2+y^2-1) \right)_y $$ where the index $y$ denotes localization (recall that in a ring $R$, if $f \in R$, then the localization $R_f$ can be constructed as the quotient $R[x]/(xf-1)$). Since the localization of an integral domain at a non-zero element is again an integral domain, it suffices to show that $x^2 + y^2 -1$ generates a prime ideal ; since $\mathbb C[x,y]$ is a UFD, it suffices to show that $x^2 + y^2 -1$ is irreducible. Can you do that?
Hope that helps,