Key idea of Pete Clark's paper on Elementary Equivalence versus Isomorphisms

 From Pop, we know that if \(K\) and \(L\) are finitely generated fields (i.e. over their prime subfields) and \( K \equiv L\) then there exist embeddings \(K \hookrightarrow L\) and \(L \hookrightarrow K\). In other words, \(K\) and \(L\) are isogeneous.  

Thus we want to see when being isogenous implies being isomorphic. Clark shows that such would be the case if we have one of the following three situations:

1. For some positive integer \(n\),  \(K\) and \(L\)  are cyclic elements of \(SB_n\), the set of function fields of Severi-Brauer varieties of dimension \(n\) over a given field \K\).

2. For some \(n \leq 2\), both \(K\) and \(L\) lie in \(Q_n\), the class of function fields of quadric hypersurfaces of dimension \(n\) over \(k\) and \( char \,k \neq 2\).

3. For some \(n > 1\), \(K \in SB_n\) and \(L \in Q_n\). 

Question: does elementarily equivalence always \( \implies\) isogeny? Observe that Pop's theorem was only for finitely generated function fields. 

Answer: If \(K \equiv L\) then their Brauer kernels are going to be Galois conjugates. Thus for any choice of \(k\)-structure on \(K\), ther is a unique \(k\) structure on \(L\) such that the Brauer kernels \(\kappa(K/k)\) and \(\kappa(L/k)\) are equal.  What Clark proof is that this is basically equivalent to \(K\) and \(L\) being isogenous, as well as to \(K\) being isomorphic to \(L\). 

The key player here is the Brauer kernel and the main idea use is the relevance of Galois cohomology and Brauer groups to the classification of Quadric surfaces.