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1 Instructions: 1. The point value of each exercise occurs adjacent to the problem. 2. No books or notes or calculators are allowed. Page Points Possible Points Total 200 1

2 1. Let G be a finite group and H a subgroup such that G : H = d with 1 < d < G. (a) (5 pts) Describe the natural homomorphism φ : G S d, where S d is the permutation group on the left cosets of H in G. (b) (5 pts) If G = n and d is the smallest prime dividing n, prove that H is normal in G. (c) (5 pts) If G = 24 and d = 3, prove that G contains a normal subgroup of order 4 or 8. (d) (5 pts) If G = 24 and d = 3, must H be normal in G? Justify your answer. 2

3 2. A sequence of subgroups 1 = N 0 N 1 N k 1 N k = G is called a composition series for a group G if N i is normal in N i+1 and N i+1 /N i is a simple group for 0 i k 1. (a) (5 pts) State the Jordan-Hölder Theorem for a finite group. (b) (5 pts) Diagram the lattice of subgroups of the symmetric group S 3 and exhibit all the composition series for S 3. How many are there? (c) ( 5 pts) Diagram the lattice of subgroups of the quaternion group Q 8 and exhibit all the composition series for Q 8. How many are there? (d) ( 5 pts) How many composition series exist for the dihedral group D 8? Justify your answer. 3

4 3. (8 pts) Let K/F be an algebraic field extension. If K = F (α) for some α K, prove that there are only finitely many subfields of K that contain F. 4. (10 pts) Let x and y be indeterminates over the field F 2. Prove that there exist infinitely many subfields of L = F 2 (x, y) that contain the field K = F 2 (x 2, y 2 ). 4

5 5. Let p be a prime integer. Recall that a field extension K/F is called a p-extension if K/F is Galois and [K : F ] is a power of p. (a) (10 pts) If K/F and L/K are p-extensions, prove that the Galois closure of L/F is a p-extension. (b) (8 pts) Give an example where K/F and L/K are p-extensions, but L/F is not Galois. 5

6 6. Let L/Q be the splitting field of the polynomial x 6 2 Q[x]. (a) (4 pts) What is the degree [L : Q]? (b) (4 pts) If α is one root of x 6 2, diagram the lattice of fields between Q and Q(α). (c) (4 pts) Give generators for each subfield K of L for which [K : Q] = 2. How many K are there? (d) (4 pts) Give generators for each subfield K of L for which [K : Q] = 3. How many K are there? (e) (4 pts) Give generators for each subfield K of L for which [K : Q] = 4. How many K are there? (f) (4 pts) How many subfields K of L are such that [L : K] = 2? 6

7 7. (9 pts) Prove that an irreducible monic polynomial f(x) Q[x] cannot have a multiple root. 8. (9 pts) Give an example of a field F having characteristic p > 0 and an irreducible monic polynomial f(x) F [x] that has a multiple root. 7

8 9. Consider the quadratic integer ring R = Z[ 5] = Z[x] (x 2 +5)Z[x]. (a) (6 pts) Give generators for each ideal of R that contains (3) and diagram the lattice of ideals of R that contain (3). (b) (6 pts) Give generators for each ideal of R that contains (7) and diagram the lattice of ideals of R that contain (7). (c) (6 pts) Including the whole ring R and the ideal (21), how many ideals of R contain the ideal (21)? (d) (6 pts) Diagram the lattice of ideals of R that contain (21). 8

9 10. (10 pts) Let n be a positive integer and d a positive integer that divides n. Suppose α R is a root of the polynomial x n 2 Q[x]. Prove that there is precisely one subfield F of Q(α) with [F : Q] = d. 11. (10 pts) Does there exist an example of an infinite abelian group G such that every proper subgroup of G is a finite group? Justify your answer with either a proof or an example. 9

10 12. Let R be the polynomial ring Z[x] (a) (9 pts) How many maximal ideals of R contain the ideal I = (15, x 2 + 2)R? Give generators for each of these maximal ideals. (b) (9 pts) Diagram the lattice of ideals of R that contain the ideal I = (15, x 2 + 2)R, giving generators for each ideal. 10

11 13. Let n and p be positive integers with p a prime integer. Let Z = x be a cyclic group of order p n 1. (a) (7 pts) Describe the group Aut(Z) of automorphism of Z. In particular, what is Aut(Z)? (b) (7 pts) Let F p be the field with p elements and let L/F p be a field extension of degree n. Let G be the Galois group of L/F p. Describe the group G. In particular, what is G? 14. (6 pts) Let G be a finite group and let C be the center of G. If G/C is abelian, does it follow that C = G? Justify your answer. 11

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