92.421/521 Abstract Algebra

Fall 2011

Problem Set #6

Due December 10

Instructions:

Do each problem on a separate sheet of paper.  On each sheet, write your name And problem statement (it can be abbreviated). Include all logical steps/observations.

1.      Let p be prime.  Prove that (Subscript[ℤ, (p)])={(a/b): a,b∈ℤ and gcd(b,p)=1} is a subring of the field of rational numbers.    Explain why (Subscript[ℤ, (p)]) is not a field.

2.     If R is a commutative ring with identity that isn’t an integral domain, then factorization of polynomials over R isn’t unique.  Illustrate this fact by showing that (x^2)+x + 8 has two different factorizations over (Subscript[ℤ, 10])[x]

3.   List all of the polynomials of degree 4 or less in (Subscript[ℤ, 2])[x] and factor them completely  From among these polynomials, identify all of the irreducible polynomials of degree 2, 3 and 4 in (Subscript[ℤ, 2])[x].

Some suggested exercises to work on, but not turn in.

Chapter 15/16
1, 5, 8

Chapter 16/17
2, 3, 19

Chapter  20/21
1d,  2 a b c h , 5