Unformatted text preview: n ³ 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 ﬁnite 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 Œ³Ł D f a³ 3 C b³ 2 C c³ 2 C d W a;b;c;d 2 Q g Š Q ŒxŁ=.p.x//; and Q Œ³Ł is a ﬁeld. Compute the inverse of ³ 2 C 1 as a polynomial in ³ . Hint: Obtain a system of linear equations for the coefﬁcients of the polynomial. 7.3.7. Let ³ D e 2±i=5 . (a) Find the minimal polynomials for cos .2´=5/ and sin .2´=5/ over Q . (b) Find the minimal polynomial for ³ over Q . cos .2´=5// ....
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
- Fall '08