7.4. SPLITTING FIELD OF A CUBIC POLYNOMIAL331Return to the irreducible cubicf .x/2KOExŁ, and suppose thatı2Kso that the splitting fieldEhas dimension 3 overK. We have seen that thegroup AutK.E/acts as permutations of the roots off .x/, and the action istransitive; that is, for any two roots˛iand˛j, there is a2AutK.E/suchthat.˛i/D˛j. However, not every permutation of the roots can arise asthe restriction of aK–automorphism ofE. In fact, any odd permutation ofthe roots would mapıD.˛1˛2/.˛1˛3/.˛2˛3/toı, so cannotarise from aK–automorphism ofE. The group of permutations of theroots induced by AutK.E/is a transitive subgroup of even permutations,so must coincide withA3. We have proved the following:Proposition 7.4.4.IfKis a subfield ofC,f .x/2KOExŁis an irreduciblecubic polynomial,Eis the splitting field off .x/inC, anddimK.E/D3,thenAutK.E/ŠA3ŠZ3.
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