Existentially generated subfields of large fields

Let L be a large field of characteristic exponent p, and let E⊆L be an infinite existentially generated subfield. We show that E contains L(pn), the pn-th powers in L, for some n<ω. This generalises a result of Fehm from [4], 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.