Lectures on the Algebraic Theory of Fields - Tata Institute of ...

3. Isomorphisms of fields 213 Isomorphisms of fieldsLet K/k be an algebraic extension of k and W any extension of K and soof k. A mappingσof K into W is said to be k− linear if forα,β∈K 24σ(α+β)=σα+σβσα∈Wand ifλ∈k,σ(λα)=λσα. Ifσis a k−linear map of K intoW we defineασ forαin W by(ασ)β=ασβforβ∈ K. This again is a k−linear map and so the k−linear maps of Kinto W form a vector space V over W.A k−isomorphismσ of K into W is obviously a k−linear map andsoσ∈V. We shall say, two isomorphismsσ,τ of K into W (trivial onk) are distinct if there exists at least oneω∈K,ω0 such thatσωτωLet S be the set of mutually distinct isomorphisms of K into W. Wethen haveTheorem 3. S is a set of linearly independent elements of V over W.Proof. We have naturally to show that every finite subset of S is linearlyindependent over W. Let on the contraryσ 1 ,...,σ n be a finite subset ofS satisfying a non trivial linear relation∑α i σ i = 0iα i ∈ W. We may clearly assume that no proper subset ofσ 1 ,...,σ nis linearly dependent. Then in the above expression allα i are differentfrom zero. Letωbe any element of K. Then∑α i σ i ω=0i□