Say we have the field extension $\Bbb{Q}[\sqrt[2]{2},\sqrt[3]{2}]$. Is this field isomorphic to $\Bbb{Q}[x,y]/(x^2-y^3)$?
I made some preliminary investigation, and this doesn't seem to be true. Is $\Bbb{Q}[\sqrt[2]{2},\sqrt[3]{2}]$ isomorphic to $\Bbb{Q}[x,y]/(x^2-y^3,x^2-2,y^3-2)$?
How should we deal with such situations?
Here's one way to show it (there are many others!). The strategy is to use the universal property of quotient rings to construct homomorphisms that can differentiate the two rings. In particular, there is a homomorphism $f:\mathbb{Q}[x,y]\to\mathbb{Q}$ which sends $x$ and $y$ to $0$. Since $f(x^2-y^3)=0$, this factors through a homomorphism $g:\mathbb{Q}[x,y]/(x^2-y^3)\to\mathbb{Q}$. On the other hand, there does not exist any homomorphism $h:\mathbb{Q}[\sqrt{2},\sqrt[3]{2}]\to\mathbb{Q}$, since $h(\sqrt{2})$ would need to be some element of $\mathbb{Q}$ whose square is $2$ and no such element exists.