We have reached the main point of this series of problems about Z p 3. Our goal is to

Chapter 39, Problem 39.28

(choose chapter or problem)

We have reached the main point of this series of problems about Z p 3. Our goal is to make a statement about unique factorization in Z p 3. Suppose we factor a into irreducibles as a D .p1/.p2/.p3/ .pt/ and consider the factorization a D .p2/.p1/.p3/ .pt/: We consider these factorizations to be the same. We do not care about the order of the factors (this is the same as for factoring positive integers into primes), and we do not care about stray factors of 1.For example, we consider the following two factorizationsof 6 into irreducibles to be the same:6 D .2/.p3/.p3/and6 D .2/.p3/.p3/:These are the same despite the fact that we use 2 in the first factorization and 2in thesecondwe do not care about sign changes in the factors.Thus the following two factorizations of 4 into irreducibles are the same:4 D .2/.2/ D .2/.2/:Here is the surprise and your job for this problem: Find another factorization of 4 intoirreducibles.Therefore, in the number system Zp3,we can factor numbers into irreducibles,but the factorization need not be unique!