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}.
Definition galois U V := [&& (U <= V)%VS, separable U V & normalField U V].

Lemma gal_independent E (P : pred (gal_of E)) (c_ : gal_of E -> L) :
(forall a, a \in E -> \sum_(x | P x) c_ x * x a = 0) ->
(forall x, P x -> c_ x = 0).