End TraceAndNormField.
Definition normalField U V := [forall x in kAEndf U, x @: V == V]%VS.

Lemma normalField_root_minPoly K E a b :
(K <= E)%VS -> normalField K E -> a \in E -> root (minPoly K a) b ->
exists2 x, x \in 'Gal(E / K) & x a = b.