College Algebra Exam Review 327

College Algebra Exam Review 327 - 337 7.5 SPLITTING FIELDS...

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Unformatted text preview: 337 7.5. SPLITTING FIELDS OF POLYNOMIALS IN C Œx E 2 2 K.˛1 / r d rr d rr 3 2 rr d r d rr d rr d K.˛2 / K.˛3 / d d 3 d 3d d d Q ¨ 3 ¨¨ 2 ¨¨ ¨ ¨ ¨ ¨¨ K.ı/ ¨¨ Figure 7.4.2. Intermediate fields for a splitting field with Galois group S3 . 7.5. Splitting Fields of Polynomials in C Œx Our goal in this section will be to state a generalization of Theorem 7.4.9 for arbitrary polynomials in KŒx for K a subfield of C and to sketch some ideas involved in the proof. This material will be treated systematically and in a more general context in the next chapter. Let K be any subfield of C , and let f .x/ 2 KŒx. According to Gauss’s fundamental theorem of algebra, f .x/ factors into linear factors in C Œx. The smallest subfield of C that contains all the roots of f .x/ in C is called the splitting field of f .x/. As for the cubic polynomial, in order to understand the structure of the splitting field, it is useful to introduce its symmetries over K . An automorphism of E is said to be a K -automorphism if it fixes every point of K . The set of K –automorphisms forms a group (Exercise 7.4.4). Here is the statement of the main theorem concerning subfields of a splitting field and the symmetries of a splitting field: Theorem 7.5.1. Suppose K is a subfield of C , f .x/ 2 KŒx, and E is the splitting field of f .x/ in C . (a) dimK .E/ D jAutK .E/j. (b) The map M 7! AutM .E/ is a bijection between the set of intermediate fields K  M  E and the set of subgroups of AutK .E/. The inverse map is H 7! Fix.H /. In particular, there are only finitely many intermediate fields between K and E . ...
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