Definition galTrace U V a := \sum_(x in 'Gal(V / U)) (x a).
Definition galNorm U V a := \prod_(x in 'Gal(V / U)) (x a).
Section TraceAndNormMorphism.
Variables U V : {vspace L}.
Fact galTrace_is_additive : additive (galTrace U V).
Canonical galTrace_additive := Additive galTrace_is_additive.

(* Specifications: *)
Lemma galNorm1 : galNorm U V 1 = 1.
Lemma galNormV : {morph galNorm U V : a / a^-1}.
Lemma galNormX n : {morph galNorm U V : a / a ^+ n}.
Lemma galNorm_prod (I : Type) (r : seq I) (P : pred I) (B : I -> L) :
galNorm U V (\prod_(i <- r | P i) B i)
= \prod_(i <- r | P i) galNorm U V (B i).