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Unformatted text preview: n log 2 . dim K .L// C 1 and there exist a 1 ;:::;a n 2 L such that L D K.a 1 ;:::;a n / . 7.3.5. Show that R is not a nite extension of Q . 7.3.6. (a) Show that the polynomial p.x/ D x 5 1 x 1 D x 4 C x 3 C x 2 C x C 1 is irreducible over Q . (b) According to Proposition 7.3.6 , Q D f a 3 C b 2 C c 2 C d W a;b;c;d 2 Q g Q x=.p.x//; and Q is a eld. Compute the inverse of 2 C 1 as a polynomial in . Hint: Obtain a system of linear equations for the coefcients of the polynomial. 7.3.7. Let D e 2i=5 . (a) Find the minimal polynomials for cos .2=5/ and sin .2=5/ over Q . (b) Find the minimal polynomial for over Q . cos .2=5// ....
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- Fall '08