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 Hilbert's_theorem_90 K E x a :
generator 'Gal(E / K) x -> a \in E ->
reflect (exists2 b, b \in E /\ b != 0 & a = b / x b) (galNorm K E a == 1).