  We establish a polynomial ergodic theorem for actions of the affine group of
a countable field $K$. As an application, we deduce--via a variant of
Furstenberg's correspondence principle--that for fields of characteristic zero,
any "large" set $E\subset K$ contains "many" patterns of the form
$\{p(x)+y,xy\}$, for every non-constant polynomial $p(x)\in K[x]$.
  Our methods are flexible enough that they allow us to recover analogous
density results in the setting of finite fields and, with the aid of a new
finitistic variant of Bergelson's "colouring trick", show that for $r\in
\mathbb{N}$ fixed, any $r-$colouring of a large enough finite field will
contain monochromatic patterns of the form $\{x,p(x)+y,xy\}$.
  In a different direction, we obtain a double ergodic theorem for actions of
the affine group of a countable field. An adaptation of the argument for affine
actions of finite fields leads to a generalisation of a theorem of Shkredov.
Finally, to highlight the utility of the aforementioned finitistic "colouring
trick", we provide a conditional, elementary generalisation of Green and
Sanders' $\{x,y,x+y,xy\}$ theorem.
