College Algebra Exam Review 321

College Algebra Exam Review 321 - 7.4 SPLITTING FIELD OF A...

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7.4. SPLITTING FIELD OF A CUBIC POLYNOMIAL 331 Return to the irreducible cubic f .x/ 2 KOExŁ , and suppose that ı 2 K so that the splitting field E has dimension 3 over K . We have seen that the group Aut K .E/ acts as permutations of the roots of f .x/ , and the action is transitive; that is, for any two roots ˛ i and ˛ j , there is a 2 Aut K .E/ such that i / D ˛ j . However, not every permutation of the roots can arise as the restriction of a K –automorphism of E . In fact, any odd permutation of the roots would map ı D 1 ˛ 2 /.˛ 1 ˛ 3 /.˛ 2 ˛ 3 / to ı , so cannot arise from a K –automorphism of E . The group of permutations of the roots induced by Aut K .E/ is a transitive subgroup of even permutations, so must coincide with A 3 . We have proved the following: Proposition 7.4.4. If K is a subfield of C , f .x/ 2 KOExŁ is an irreducible cubic polynomial, E is the splitting field of f .x/ in C , and dim K .E/ D 3 , then Aut K .E/ Š A 3 Š Z 3 .
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