Unformatted text preview: 328 7. FIELD EXTENSIONS – FIRST LOOK 7.3.8. Show that the splitting field E of x 3 2 has dimension 6 over Q . Refer to Exercise 7.2.8 . 7.3.9. If ˛ is a fifth root of 2 and ˇ is a seventh root of 3, what is the dimension of Q .˛;ˇ/ over Q ? 7.3.10. Find dim Q Q .˛;ˇ/ , where (a) ˛ 3 D 2 and ˇ 2 D 2 (b) ˛ 3 D 2 and ˇ 2 D 3 7.3.11. Show that f.x/ D x 4 C x 3 C x 2 C 1 is irreducible in Q OExŁ . For p.x/ 2 QOExŁ , let p.x/ denote the image of p.x/ in the field QOExŁ=.f.x// . Compute the inverse of . x 2 C 1/ . 7.3.12. Show that f.x/ D x 3 C 6x 2 12x C 3 is irreducible over Q . Let be a real root of f.x/ , which exists due to the intermediate value theorem. Q . / consists of elements of the form a C a 1 C a 2 2 . Explain how to compute the product in this field and find the product .7 C 2 C 2 /.1 C 2 / . Find the inverse of .7 C 2 C 2 / . 7.3.13. Show that f.x/ D x 5 C 4x 2 2x C 2 is irreducible over Q . Let be a real root of f.x/ , which exists due to the intermediate value theorem., which exists due to the intermediate value theorem....
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 Fall '08
 EVERAGE
 Algebra, Intermediate Value Theorem, KŒx

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