College Algebra Exam Review 317

College Algebra Exam Review 317 - n ³ log 2 dim K.L C 1...

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7.3. ADJOINING ALGEBRAIC ELEMENTS TO A FIELD 327 We have already observed that the set of algebraic numbers is count- able, so this set is a countable field. n Exercises 7.3 7.3.1. Show that if K ± L are fields, then the identity of K is also the identity of L . Conclude that L is a vector space over K . 7.3.2. Fill in the details of the proof of 7.3.1 to show that f ± i ² j g spans M over K . 7.3.3. If K ± L is a finite field extension, then there exist finitely many elements a 1 ;:::;a n 2 L such that L D K.a 1 ;:::;a n /: Give a different proof of this assertion as follows: If the assertion is false, show that there exists an infinite sequence a 1 ;a 2 ;::: of elements of L such that K ² ¤ K.a 1 / ² ¤ K.a 1 ;a 2 / ² ¤ K.a 1 ;a 2 ;a 3 / ² ¤ :::: Show that this contradicts the finiteness of dim K .L/ . 7.3.4. Suppose dim K .L/ < 1 . Show that there exists a natural number n such that
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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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