Page 229 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra Sachin Kaushal, Lovely Professional University

Notes Unit 23: Division of Algorithm

CONTENTS
Objectives

Introduction
23.1 The Division Algorithm
23.2 Summary
23.3 Keywords
23.4 Review Questions

23.5 Further Readings

Objectives

After studying this unit, you will be able to:
Prove and use the division algorithm for F[X], where F is a field

Discuss examples related to algorithms

Introduction

In the last unit, you have studied about polynomials rings. In this unit, we will discuss the
division of algorithm.

23.1 The Division Algorithm

We have discussed various properties of divisibility in Z. In particular, we proved the division
algorithm for integers. We will now do the same for polynomials over a field F.
Theorem 1 (Division Algorithm): Let F be a field. Let f(x) and g(x) be two polynomials in F[x],
with g(x)  0. Then

(a) there exist two polynomials q(x) and r (x) in F [X] such that
f (x) = q (x) g (x) + r (x), where deg r(x) < deg g (x).
(b) the polynomials q (x) and r (x) are unique.

Proof: (a) If deg f (x) < deg g (x), we can choose q (x) = 0.
Then f(x) = 0. g(x) + f (x), where deg f(x) < deg g (x).
Now, let us assume that deg f(x)  deg g (x).
Let f(x) = a + a x + . . . +a x , a,  0, and
n
1
n
0
g(x) = b + b x + ... + b x , b  0, with n  m.
m
m
0
1
m

222 LOVELY PROFESSIONAL UNIVERSITY