College Algebra Exam Review 330

College Algebra Exam Review 330 - fundamental theorem of...

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340 7. FIELD EXTENSIONS – FIRST LOOK (b) For all ˛ 2 E , the minimal polynomial of ˛ over K splits into linear factors over E . (c) E is the splitting field of a polynomial in KŒxŁ . Corollary 7.5.9. If K ± E ± C and E is Galois over K , then E is Galois over every intermediate field K ± M ± E . Thus far we have sketched “half” of Theorem 7.5.1 , namely, the map from intermediate fields M to subgroups of Aut K .E/ , M 7! Aut M .E/ , is injective, since M D Fix . Aut M .E// . It remains to show that this map is surjective. The key to this result is the equality of the order of subgroup with the dimension of E over its fixed field: If H is a subgroup of Aut K .E/ and F D Fix .H/ , then dim F .E/ D j H j : (7.5.1) The details of the proof of the equality ( 7.5.1 ) will be given in Section 9.5 , in a more general setting. Now, consider a subgroup H of Aut K .E/ , let F be its fixed field, and let H be Aut F .E/ . Then we have H ± H , and Fix . H/ D Fix . Aut F .E// D F D Fix .H/: By the equality ( 7.5.1 ) j H j D dim F .E/ D j H j , so H D H . This shows that the map M 7! Aut M .E/ has as its range all subgroups of Aut K .E/ . This completes the sketch of the proof of Theorem 7.5.1 , which is known as the
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Unformatted text preview: fundamental theorem of Galois theory . Example 7.5.10. The field E D Q . p 2; p 3/ is the splitting field of the polynomial f.x/ D .x 2 ² 2/.x 2 ² 3/ , whose roots are ˙ p 2; ˙ p 3 . The Galois group G D Aut Q .E/ is isomorphic to Z 2 ³ Z 2 ; G D f e;˛;ˇ;˛ˇ g , where ˛ sends p 2 to its opposite and fixes p 3 , and ˇ sends p 3 to its op-posite and fixes p 2 . The subgroups of G are three copies of Z 2 generated by ˛ , ˇ , and ˛ˇ . Therefore, there are also exactly three intermediate fields between Q and E , which are the fixed fields of ˛ , ˇ , and ˛ˇ . The fixed field of ˛ is Q . p 3/ , the fixed field of ˇ is Q . p 2/ , and the fixed field of ˛ˇ is Q. p 6/ . Figure 7.5.1 shows the lattice of intermediate fields, with the dimensions of the extensions indicated. There is a second part of the fundamental theorem, which describes the special role of normal subgroups of Aut K .E/ ....
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