- #1
evinda
Gold Member
MHB
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Hi! (Smile)
Can I justify like that, that $C[x,y,z]/<y-x^2,z-x^3>$ is an integral domain?
We show that $ker(\phi)=<y-x^2,z-x^3>$.
From the theorem of isomorphism, we have that $C[x,y,z]/<y-x^2,z-x^3> \cong im(\phi)$
$im(\phi)$ is a subring of $\mathbb{C}[x]$
$\mathbb{C}[x]$ is an integral domain, since $\mathbb{C}$ is a field.
As $im(\phi)$ is a subring of $\mathbb{C}[x]$, $im(\phi)$ is also an integral domain.
(Thinking)
Can I justify like that, that $C[x,y,z]/<y-x^2,z-x^3>$ is an integral domain?
We show that $ker(\phi)=<y-x^2,z-x^3>$.
From the theorem of isomorphism, we have that $C[x,y,z]/<y-x^2,z-x^3> \cong im(\phi)$
$im(\phi)$ is a subring of $\mathbb{C}[x]$
$\mathbb{C}[x]$ is an integral domain, since $\mathbb{C}$ is a field.
As $im(\phi)$ is a subring of $\mathbb{C}[x]$, $im(\phi)$ is also an integral domain.
(Thinking)
