# Testing isomorphism of finitely generated algebras

Let $A=\mathbf{Q}[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over the rational numbers. Let $B=\mathbf{Q}[f_1,\ldots,f_r]$ and $C=\mathbf{Q}[g_1,\ldots,g_s]$ be two finitely generated $\mathbf{Q}$-subalgebras of $A$ with explicit generators.

Q1: Is there a finite time (efficient) algorithm that allows one to say when is $B\simeq C$ as $\mathbf{Q}$-algebra?

Q2: Is there a finite time (efficient) algorithm that allows one to say when is $Frac(B)\simeq Frac(C)$?

Here $Frac(B)$ denotes the fraction field. In both questions I really mean isomorphic and not equal.

• I don't know much about algorithmic questions like this. But: it would probably help to specify whether you are given a set of generators for $B$ and $C$, or whether you merely have some implicit description of these subalgebras. – MTS Jun 9 '12 at 0:06
• Since $B$ and $C$ are both subalgebras of the same ring $A$, do you want to know whether they are isomorphic, or whether they are equal? – David E Speyer Jun 9 '12 at 0:52
• @David, here I really mean isomorphic. Equal would be "easy" since one may compute the relation ideal for $B$ and $C$ and then test for equality. – Hugo Chapdelaine Jun 9 '12 at 2:42
• @MTS, yes in both cases I have explicit sets of generators. – Hugo Chapdelaine Jun 9 '12 at 2:46
• mathoverflow.net/questions/21883/… – M P Jun 9 '12 at 3:18