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Sachin Kaushal, Lovely Professional University Unit 24: Irreducibility and Field Extensions

Unit 24: Irreducibility and Field Extensions Notes

CONTENTS
Objectives

Introduction
24.1 Irreducibility in Q[x]
24.2 Field Extensions
24.2.1 Prime Fields
24.2.2 Finite Fields

24.3 Summary
24.4 Keywords
24.5 Review Questions

24.6 Further Readings

Objectives

After studying this unit, you will be able to:
Prove and use Eisenstein’s criterion for irreducibility in Z[x] and Q[x]

Obtain field extensions of a field F from F[x]

Obtain the prime field of any field

Use the fact that any finite field F has pn elements, where char F = p and dim z F = n
 p
Introduction

We have discussed various kinds of integral domains, including unique factorisation domains.
Over there you saw that Z[x] and Q[x] are UFDs. Thus, the prime and irreducible elements
coincide in these rings. In this unit, we will give you a method for obtaining the prime
(or irreducible) elements of Z[x] and Q[x]. This is the Eisenstein criterion, which can also be used
for obtaining the irreducible elements of any polynomial ring over a UFD.
After this, we will introduce you to the field extensions and subfields. We will use irreducible
polynomials for obtaining field extensions of a field F from F[x]. We will also show you that
every field is a field extension of Q or Z, for some prime p. Because of this, we call Q and the Z s
p
prime fields. We will discuss these fields briefly.
Finally, we will look at finite fields. These fields were introduced by the young French
mathematician Evariste Galois while he was exploring number theory. We will discuss some
properties of finite fields which will show us how to classify them.
Before reading this unit ,we suggest that you go through the definitions of irreducibility.

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