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

Lemma normalField_factors K E :
(K <= E)%VS ->
reflect {in E, forall a, exists2 r : seq (gal_of E),
r \subset 'Gal(E / K)
& minPoly K a = \prod_(x <- r) ('X - (x a)%:P)}
(normalField K E).