# On an isomorphism of rings

Let $$k$$ be a field, it's claimed in algebraic geometry textbook that $$k[v^2, v^3]\cong k[t, u]/(t^2-u^3)$$ via $$v^2\mapsto u, v^3\mapsto t$$. But I can't show it's well-defined, since an integer can have many ways to write as $$2i+3j$$. Any idea?

• Yes, for instance $20=2\times1+3\times 6$ and $20=2\times7+3\times 2$. So $v^{20}$ should go to $ut^6$ and $u^7t^2$? Luckily that's not a problem, as $ut^6-u^7t^2$ is a multiple of $t^2-u^3$. Jun 26, 2020 at 14:41
• OK, I think I know how to do it. Jun 26, 2020 at 15:03

Instead you should try to show that $$\phi : k[t,u] \longrightarrow k[v^2,v^3]$$ has kernel generated by $$(t^2-u^3).$$ This map will be surjective, and well defined.The first isomorphism theorem will give you the desired result.
• You don't need to worry about using induction. What you should take $\phi$ as described, and ask yourself if you had a polynomial $\phi(p(t,u))=0,$ that is $p(v^2,v^3)=0,$ then what could you say about the kernel of this map? Angina Seng gave you the way to see the answer in their comment on your original post. Jun 26, 2020 at 14:47