(e) Let G = S4 and H = {e, (1 2)}. What is the size of NG(H), the normalizer of H in G? (f) What is the characteristic of the ring Z/3ZXZ/2Z? (g) Ifo: Z/14ZZ/10Z is a ring homomorphism, then what are the possibilities for |ker()|?

Write 20 as the direct sum of two of its nontrivial subgroups.

12. Consider the mapping defined by . Decide whether is a homomorphism, and justify your decision.

Let R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)

Prove Corollary 8.18: A polynomial of positive degree over the field has at most distinct zeros in

11. a. Give an example of a ring of characteristic 4, and elements in such that b. Give an example of a noncommutative ring with characteristic 4, and elements in such that .

Let R be a commutative ring with characteristic 2. Show that each of the following is true for all x,yR a. (x+y)2=x2+y2 b. (x+y)4=x4+y4

Let :312 be defined by ([x]3)=4[x]12 using the same notational convention as in Exercise 9. Prove that is a ring homomorphism. Is (e)=e where e is the unity in 3 and e is the unity in 12?

18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .

Examples 5 and 6 of Section 5.1 showed that P(U) is a commutative ring with unity. In Exercises 4 and 5, let U={a,b}. Is P(U) a field? If not, find all nonzero elements that do not have multiplicative inverses. [Type here][Type here]