  We study subfields of large fields which are generated by infinite
existentially definable subsets. We say that such subfields are existentially
generated.
  Let $L$ be a large field of characteristic exponent $p$, and let $E\subseteq
L$ be an infinite existentially generated subfield. We show that $E$ contains
$L^{(p^{n})}$, the $p^{n}$-th powers in $L$, for some $n<\omega$. This
generalises a result of Fehm, which shows $E=L$ under the assumption that $L$
is perfect. Our method is to first study existentially generated subfields of
henselian fields. Since $L$ is existentially closed in the henselian field
$L((t))$, our result follows.
