Page 220 - DMTH403_ABSTRACT_ALGEBRA

Richa Nandra, Lovely Professional University Unit 22: Polynomial Rings

Unit 22: Polynomial Rings Notes

CONTENTS
Objectives

Introduction
22.1 Ring of Polynomials
22.2 Some Properties of R[x]
22.3 Summary
22.4 Keywords

22.5 Review Questions
22.6 Further Readings

Objectives

After studying this unit, you will be able to:
Identify polynomials over a given ring

Prove and use the fact that R [x], the set of polynomials over a ring R, is a ring

Relate certain properties of R[x] to those of R

Introduction

In the earlier units, you must have come across expressions of the form x+1, x +2x+1, and so on.
2
These are examples of polynomial. You have also dealt with polynomials in the course of Linear
Algebra. In this unit, we will discuss sets whose elements are polynomials of the type a + a, x +
0
... + a x , where a , a ,......, a,, are elements of a ring R. You will see that this set, denoted by R [x],
n
n
0
1
is a ring also.
You may wonder why we are talking of polynomial rings in a block on domains and fields. The
reason for this is that we want to focus on a particular case, namely, R [x], where R is a domain.
This will turn out to be a domain also, with a lot of useful properties. In particular, the ring of
polynomials over a field satisfies a division algorithm, which is similar to the one satisfied by
Z. We will prove this property and use it to show how many roots any polynomial over a field
can have.
22.1 Ring of Polynomials

As we have said above, you may already be familiar with expressions of the type 1 + x, 2 + 3x +
4x , x -1, and so on. These are examples of polynomials over the ring Z. Do these examples
2
5
suggest to you what a polynomial over any ring R is ? Let’s hope that your definition agrees
with the following one.
Definition: A polynomial over a ring R in the indeterminate x is an expression of the form

a x + a x + a x + ... + a x ,
0
n
2
1
0 1 2 n
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