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Unformatted text preview: fundamental theorem of Galois theory . Example 7.5.10. The eld E D Q . p 2; p 3/ is the splitting eld of the polynomial f.x/ D .x 2 2/.x 2 3/ , whose roots are p 2; p 3 . The Galois group G D Aut Q .E/ is isomorphic to Z 2 Z 2 ; G D f e;;; g , where sends p 2 to its opposite and xes p 3 , and sends p 3 to its opposite and xes p 2 . The subgroups of G are three copies of Z 2 generated by , , and . Therefore, there are also exactly three intermediate elds between Q and E , which are the xed elds of , , and . The xed eld of is Q . p 3/ , the xed eld of is Q . p 2/ , and the xed eld of is Q. p 6/ . Figure 7.5.1 shows the lattice of intermediate elds, 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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 Fall '08
 EVERAGE
 Algebra, Factors

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