End Matrix.

(* Specifications: *)
Lemma dim_fixedField E (G : {group gal_of E}) : #|G| = \dim_(fixedField G) E.
Lemma dim_fixed_galois K E (G : {group gal_of E}) :
galois K E -> G \subset 'Gal(E / K) ->
\dim_K (fixedField G) = #|'Gal(E / K) : G|.
Lemma gal_fixedField E (G : {group gal_of E}): 'Gal(E / fixedField G) = G.
Lemma gal_generated E (A : {set gal_of E}) : 'Gal(E / fixedField A) = <<A>>.
Lemma fixedField_galois E (A : {set gal_of E}): galois (fixedField A) E.
Section FundamentalTheoremOfGaloisTheory.