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Unformatted text preview: 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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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra, Permutations

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