PH. D. COMPREHENSIVE EXAM ALGEBRA JANUARY, 1993

1. Pair each group in column A with an isomorphic group in column B. Here, Z_{n} = Z/nZ, Z_{n}^{×} = group of units with respect to multiplication in Z_{n}, S_{n} = symmetric group on n objects, A_{n} = alternating group on n objects.

A  B
Z_{6} × Z_{2}  Z_{15} × Z_{12}^{×}
Z_{15}^{×}  Z_{5}^{×} × (263Z/526Z)
Z_{30} × Z_{2} Z_{2} × A_{3} × (S_{20}/A_{20})
Z_{8}^{×} × Z_{15}  Z_{11}^{×} × Z_{7}^{×}
Z_{8}  Z_{64}/8Z_{64}

2. Prove or disprove: Every group of order 27 is abelian.

3. Prove or disprove: Z[x] is a principal ideal domain.

4. Let R be a ring with a two-sided ideal I. Suppose there exists e ∈ I such that ei = ie = i for all i ∈ I.

(a) Show that e lies in the center of R.

(b) Show that there exists a two-sided ideal J of R such that R = I ⊕J.

5. Factor the polynomial x^{5} − 1 completely into irreducible factors over the following fields and justify your answer: (a) Q (b) R (c) C (d) Z/11Z (e) Z/13Z.

6. In each of parts (a) through (e) of problem 5, compute the Galois group of the splitting field of x^{5} − 1 over the indicated field and justify your answer.