MATH 241 Final, Fall 1996

eee fis Bie MN American University of Beirut Math. gui A. Shamsuddin Final Exam 4 February, 1997 (1) Define the following terms. (i) Group. (ii) Subgroup. (iii) Normal subgroup. (iv) Index of a subgroup in a group. (v) Group homomorphism. (vi) Ring. (vii) Ring homomorphism. (viti) Ideal of a ring. (ix) Principal ideal domain. (x) Field. (xi) Characteristic of a field. (2) State, without proof, the following theorems. (i) Cayley’s theorem. (ii) Cauchy’s theorem. (iti) The first part of Sylow’s theorem. (3) How many automorphisms does the group Z143 have? (4) Find the remainder of dividing 100003!° by 143. A stupid method, such as using the calculator, is worse than a wrong answer. (5) Show that if G is a group and a,6€ G, then (ab)~! = ba’. (6) State and prove Lagrange’s theorem. (7) Show that if & is a finite commutative integral domain, then R is a field. (8) Define the field of fractions of a commutative integral domain. Carry out the construction of the field of fractions, without showing the details of the proof. (9) Give an example of a division ring which is not commutative. Remember the quaternions? (10) Let R be a ring and let e € R be an element for which e = e?. Show that (ex — ere) = (ne — exe)? =0 for all zc € R. Deduce that if R is a ring in which z? = z for all xz € R then R is commutative.