College Algebra Exam Review 324

College Algebra Exam Review 324 - 334 7. FIELD EXTENSIONS...

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Unformatted text preview: 334 7. FIELD EXTENSIONS – FIRST LOOK the proof, it suffices to consider the case that K M ¤ H ¤ ¤ E and fe g ¤ S3 . Then we have dimE .M / 2 f2; 3g, and M  M  E , so either M D M , or M D E . In Exercise 7.4.7, the fixed point subfield is computed for each subgroup of AutK .E/ Š S3 . The result is Fix.Hi / D K.˛i /, Fix.A3 / D K.ı/ [and, of course, Fix.S3 / D K and Fix.fe g/ D E ]. Consequently, if H D Hi , then M D K.˛i / ¤ E , so M D M D K.˛i /. Similarly, if H D A3 , then M D K.ı/ ¤ E , so M D M D K.ı/. I We have proved the following: Theorem 7.4.9. Let K be a subfield of C , let f .x/ 2 KŒx be an irreducible cubic polynomial, and let E be the splitting field of f .x/ in C . Then there is a bijection between subgroups of AutK .E/ and intermediate fields K  M  E . Under the bijection, a subgroup H corresponds to the intermediate field Fix.H /, and an intermediate field M corresponds to the subgroup AutM .E/. This is a remarkable result, even for the cubic polynomial. The splitting field E is an infinite set. For any subset S  E , we can form the intermediate field K.S /. It is certainly not evident that there are at most six possibilities for K.S / and, without introducing symmetries, this fact would remain obscure. pp 3/ E D Q. 3 2; 2 p Q. 3 2/ r d rr d rr 3 2 2 rr d r d rr d rr dp p p 3 3 Q.! 2/ Q.! 2 d d 3 d 3d d d Q ¨ 3 ¨¨ 2 ¨¨ ¨ ¨ 2/ ¨ ¨¨ Q. 3/ ¨¨ Figure 7.4.1. Intermediate fields for the splitting field of f .x/ D x 3 2 over Q. ...
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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.

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