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Name: Problem Set 4 Math 416, Section 200, Spring 2014 Due: Thursday, February 20th. Review Sections 33, 34, 35 in your textbook. Complete the following items, staple this page to the front of your work, and turn your assignment in at the beginning of class on Thursday, February 20th. Remember to fully justify all your answers, and provide complete details. 1. Show that any two fields of order pn (where p is prime) are isomorphic. 2. Find the number of primitive roots of unity in GF(9). 3. Read the proof of Theorem 34.2. Then do Exercise 34.1. 4. Read the proof of Theorem 34.5. Then do Exercise 34.3. 5. Read the proof of Theorem 34.7. Then do Exercise 34.5. 6. Let K and L be normal subgroups of a group G, and assume that K ∨ L = G and K ∩ L = {e}. Show that G/K L and G/L K. 7. Find all composition series of Z60 and show that they are isomorphic. 8. a. Find the center of S 3 × Z4 , that is, find Z(S 3 × Z4 ) = {z ∈ S 3 × Z4 | zg = gz for all g ∈ S 3 × Z4 }. b. Read the definition of ascending central series (Definition 35.20). Find the ascending central series of S 3 × Z4 . 9. Show that if H0 = {e} < H1 < H2 < · · · < Hn = G is a subnormal (or normal) series for a group G, and if Hi+1 /Hi is of finite order si+1 , then G is of finite order s1 s2 · · · sn . 10. Show that a subgroup of a solvable group is solvable. 11. Extra Credit. a. Show that if f (x) is an irreducible polynomial in Z p [x], there exists n ∈ Z+ such that f (x) n is a divisor of x p − x. Page 1 n b. Show that x p − x is the product of all monic irreducible polynomials in Z p [x] whose degree divides n. Through the course of this assignment, I have followed the Aggie Code of Honor. An Aggie does not lie, cheat or steal or tolerate those who do. Signed: Page 2