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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 is a seventh root of 3, what is the dimension of Q .;/ over Q ? 7.3.10. Find dim Q Q .;/ , where (a) 3 D 2 and 2 D 2 (b) 3 D 2 and 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 OEx . For p.x/ 2 QOEx , let p.x/ denote the image of p.x/ in the field QOEx=.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