# Algorithm to decide whether two quotient rings are isomorphic?

Let $$I_1,I_2\subset \mathbb{C}[x_1,\dots,x_n]=:R$$ be two prime ideals (in practice we would probably work with $$\mathbb{Q}$$ instead). Is there an algorithm to decide whether $$R/I_1\cong R/I_2$$ as $$\mathbb{C}$$-algebra's? i.e. whether $$I_1$$ and $$I_2$$ defined isomorphic varieties?

• And probably $I_1$ and $I_2$ are input by a finite generating family.
– YCor
Apr 9, 2019 at 21:26
• @YCor Yes exactly. Apr 9, 2019 at 21:27
• Note that if you input finite sets $F_1,F_2$ of rational polynomials, the questions whether $\mathbf{Q}[x_1,\dots,x_n]/\langle F_1\rangle$ and $\mathbf{Q}[x_1,\dots,x_n]/\langle F_2\rangle$ are isomorphic, resp. whether $\mathbf{C}[x_1,\dots,x_n]/\langle F_1\rangle$ and $\mathbf{C}[x_1,\dots,x_n]/\langle F_2\rangle$ are isomorphic, are different.
– YCor
Apr 10, 2019 at 8:42
• @YCor Yes I realise that, but I'm curious about both questions. To point is that I'm actually most interested in the question over $\mathbb{C}$, but there is no hope of this being actually computable, hence the question over $\mathbb{Q}$, where more things do tend to be computable. Actually I guess we could work over $\overline{\mathbb{Q}}$. Apr 10, 2019 at 8:47
• You can work over an algebraically closed computable field (computable means that you can input its elements, compute operations, and determine whether two elements are equal).
– YCor
Apr 10, 2019 at 8:52

Macaulay2 has an algorithm to compute minimal presentations of $$I_1$$ and $$I_2$$ ("compute a minimal presentation of the quotient ring defined by an ideal"). If those minimal presentation are equal (or otherwise observably isomorphic), then $$R/I_1\cong R/I_2$$.