Page 286 - DMTH403_ABSTRACT_ALGEBRA

Richa Nandra, Lovely Professional University Unit 31: The Galois Group of a Polynomial

Unit 31: The Galois Group of a Polynomial Notes

CONTENTS
Objectives

Introduction
31.1 Galois Group of Polynomial
31.2 Fundamental Theorem of Symmetric Functions
31.3 Symmetric Polynomial
31.4 Constructible Polygon

31.5 Connection to Pascal’s Triangle
31.6 Summary
31.7 Keywords

31.8 Review Questions
31.9 Further Readings

Objectives

After studying this unit, you will be able to:
Discuss the Galois group of polynomial

Explain the theorem of Galois theory

Introduction

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 automorphisms determined by these permutations. We note that any automorphism
of a field F must leave its prime subfield fixed.

31.1 Galois Group of Polynomial

Proposition: Let F be an extension field of K. The set of all automorphisms  : F > F such that (a)
= a for all a in K is a group under composition of functions.
Definition: Let F be an extension field of K. The set

{ in Aut(F) | (a) = a for all a in K }
is called the Galois group of F over K, denoted by Gal(F/K).
Definition: Let K be a field, let f(x) be a polynomial in K[x], and let F be a splitting field for f(x)
over K. Then Gal(F/K) is called the Galois group of f(x) over K, or the Galois group of the
equation f(x) = 0 over K.

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