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

2. Normal extensions 19for all automorphismsσ ofΩ/k. We shall call such fields, normal extensionsof k inΩ.Let K/k be a normal extension of k andΩalgebraic closure of kcontaining K. Letα∈Kand C α the class ofα. We assert that C α ⊂ K.For ifβis an element of C α , there is an automorphismσ ofΩ/k forwhichβ=σα. SinceσK⊂K, it follows thatβ∈K. Now any elementαin K is a root of an irreducible polynomial in k[x]. Since all the elementsof C α are roots of this polynomial, it follows that if f (x) is an irreduciblepolynomial with one root in K, then all roots of f (x) lie in K.Conversely let K be a subfield ofΩ/k with this property. Letσbean automorphism ofΩ/k andα/K. Letσbe an automorphism ofΩ/kandα∈K. Let C α be the class ofα. Since C α ⊂ K,σα∈K. Butαisarbitrary in K. ThereforeσK⊂ Kand K is normal. Thus theTheorem 2. Let k⊂K⊂Ω. ThenσK= K for all automorphismsσofΩ/k⇐⇒ every irreducible polynomial f (x)∈k[x] which has one rootin K has all roots in K.Let f (x) be a polynomial in k[x] and K its splitting field. LetΩbean algebraic closure of K. Letα 1 ,...,α n be the distinct roots of f (x) in 22Ω. ThenK= k(α 1 ,...,α n )Letσbe an automorphism ofΩ/k.σα j =α j for some j. Thusσtakes the setα 1 ,...,α n onto itself. Since every element of K is a rationalfunction ofα 1 ,...,α n , it follows thatσK⊂ K. Thusi) The splitting field of a polynomial in k[x] is a normal extension ofk.Let{K α } be a family of normal subfields ofΩ/k. Then ⋂ K α isαtrivially normal. Consider k( ⋃ K α ). This again is normal since forαany automorphismσ ofΩ/k.⎛⋃ ⋃ ⋃σk⎜⎝K α⎞⎟⎠⎛⎜⎝ ⊂ k σK α⎞⎟⎠⎛⎜⎝ ⊂ k K α⎞⎟⎠ααα