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We establish a "diagonal" ergodic theorem involving the additive andmultiplicative groups of a countable field $K$ and, with the help of a newvariant of Furstenberg's correspondence principle, prove that any "large" setin $K$ contains many configurations of the form $\{x+y,xy\}$. We also show thatfor any finite coloring of $K$ there are many $x,y\in K$ such that $x,x+y$ and$xy$ have the same color. Finally, by utilizing a finitistic version of ourmain ergodic theorem, we obtain combinatorial results pertaining to finitefields. In particular we obtain an alternative proof for a result obtained byCilleruelo [11], showing that for any finite field $F$ and any subsets$E_1,E_2\subset F$ with $|E_1||E_2|>6|F|$, there exist $u,v\in F$ such that$u+v\in E_1$ and $uv\in E_2$.

Ergodic Theorem involving additive and multiplicative groups of a field and $\{x+y,xy\}$ patterns