College Algebra Exam Review 332

College Algebra Exam Review 332 - 342 7 FIELD EXTENSIONS...

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342 7. FIELD EXTENSIONS – FIRST LOOK As the size of the Galois group is also 8, each of these assignments must determine an element of the Galois group. In particular, we single out two Q –automorphisms of E : W 4 p 2 7! i 4 p 2; i 7! i; and W 4 p 2 7! 4 p 2; i 7! i: The automorphism is complex conjugation restricted to E . Evidently, is of order 4, and is of order 2. Furthermore, we can compute that D 1 . It follows that the Galois group is generated by and and is isomorphic to the dihedral group D 4 . You are asked to check this in the Exercises. We identify the Galois group D 4 as a subgroup of S 4 , acting on the roots ˛ r D i r 1 4 p 2 , 1 r 4 . With this identification, D .1234/ and D .24/ . D 4 has 10 subgroups (including D 4
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