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Name: Problem Set 4 Math 416, Section 500, 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. 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 }. 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 the direct product of two solvable groups is solvable. 11. Extra Credit. Let Z2 be the algebraic closure of Z2 . Let α ∈ Z2 be a zero of x3 + x2 + 1 and β ∈ Z2 be a zero of x3 + x + 1. Show (using the results of Section 33) that Z2 (α) = Z2 (β). Page 1 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