Section TraceAndNormField.
Variables K E : {subfield L}.
End TraceAndNormField.
Definition normalField U V := [forall x in kAEndf U, x @: V == V]%VS.

Lemma normalFieldP K E :
reflect {in E, forall a, exists2 r,
all (mem E) r & minPoly K a = \prod_(b <- r) ('X - b%:P)}
(normalField K E).