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

4. Algebraic Closure 9thatτ·σ is a homomorphism of k[x] on k ′ [x]/(F σ (x)). The kernel of thehomomorphism is the set ofϕ(x) in k[x] such that( )ϕ σ (x)∈ f σ (x).This set is precisely ( f (x)). Thus10k[x]/( f (x))≃k ′ [x]/( f ¯σ (x))By our identification, the above fields contain k and k ′ respectivelyas subfields so that there is an isomorphismµof k(α) on k ′ (β) and therestriction ofµto k isσ.In particular if k=k ′ , then k(α) and k(β) are k− isomorphic i.e., theyare isomorphic by means of an isomorphism which is identity on k. Wehave thereforeTheorem 4. If f (x)∈k[x] is irreducible andαandβare two roots ofit (either in the same extension field of k or in different extension fields),k(α) and k(β) are k− isomorphic.Note that the above theorem is false if f (x) is not irreducible in k[x].4 Algebraic ClosureWe have proved that every polynomial over k has a splitting field. For agiven polynomial this field might very well coincide with k itself. Supposek has the property that every polynomial in k has a root in k. Thenit follows that the only irreducible polynomials over k are linear polynomials.We make now theDefinition. A fieldΩis algebraically closed if the only irreducible polynomialsinΩ[x] are linear polynomials.We had already defined the algebraic closure of a field k containedin a field K. Let us now make theDefinition. A fieldΩ/k is said to be an algebraic closure of k if