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Let K be the fraction field of C[x,y] where C denotes the complex numbers.

Suppose D and E are two central division algebras over K of degree n, i.e. dim(D)=dim(E)=n^2. Is there any natural criterium to say when D and E are isomorphic as division algebra over K ?

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Dear Albert, the key theoretical tool in your problem is the theorem (due independently to Auslander-Brumer and Fadeev) relating the Brauer group of a field $k$ and that of its rational function field $K=k(y)$.Let me state the formula: $$Br(K)= Br(k)\oplus (\oplus_ {f\in P} X( k_f)) $$ Explanation The set of irreducible monic polynomials $f=f(t)\in k[t]$ has been denoted $P$, and for each $f\in P$ the field $k[t]/(f(t))$ has been denoted $k_f$. Finally, in the formula the notation $X(L)$ for a field $L$ means the group of continuous morphisms $Gal(\bar L /L) \to\mathbb Q/\mathbb Z$ (where the Galois group has the profinite topology and $\mathbb Q/\mathbb Z$ the discrete one)

Application in your case things are pretty simple since the Brauer group of $k=\mathbb C (x)$ is zero by Tsen's theorem. So you get the formula $$Br(\mathbb C(x,y))= \oplus_ {f\in P} X( \mathbb C(x)_f) $$ The final answer to your question is then that your division algebras are isomorphic if and only if their classes are equal in $Br(\mathbb C(x,y))$. Of course depending on how explicitly these division algebras are given, you might have some work to do in order to carefully follow through the isomorphism given by Auslander-Brumer and Fadeev.

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