Page 270 - DMTH403_ABSTRACT_ALGEBRA

Sachin Kaushal, Lovely Professional University Unit 28: Galois Theory

Unit 28: Galois Theory Notes

CONTENTS
Objectives

Introduction
28.1 Galois Theory
28.2 Repeated Roots
28.3 The Fundamental Theorem
28.4 Summary

28.5 Keywords
28.6 Review Questions
28.7 Further Readings

Objectives

After studying this unit, you will be able to:
Discuss Galois theory

Describe repeated roots

Introduction

In the last unit, you have studied about extension field. This unit will provide information
related to Galois theory.
28.1 Galois Theory

This gives the definition of the Galois group and some results that follow immediately from the
definition. We can give the full story for Galois groups of finite fields.

We use the notation Aut(F) for the group of all automorphisms of F, that is, all one-to-one
functions from F onto F that preserve addition and multiplication. The smallest subfield containing
the identity element 1 is called the prime subfield of F. If F has characteristic zero, then its prime
subfield is isomorphic to Q, and if F has characteristic p, for some prime number p, then its
prime subfield is isomorphic to Z . In either case, for any automorphisms  of F we must have
p
(x) = x for all elements in the prime subfield of F.
To study solvability by radicals of a polynomial equation f(x) = 0, we let K be the field generated
by the coefficients of f(x), and let F be a splitting field for f(x) over K. Galois considered
permutations of the roots that leave the coefficient field fixed. The modern approach is to
consider the automorphism determined by these permutations. The first result is that if F is an
extension field of K, then the set of all automorphism  : F  F such that (a) = a for all a  K is
a group under composition of functions. This justifies the following definitions.

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