Page 297 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra Richa Nandra, Lovely Professional University

Notes Unit 32: Solvability by Radicals

CONTENTS
Objectives

Introduction
32.1 Radical Extension
32.2 Solvable by Radicals
32.3 Summary
32.4 Keywords

32.5 Review Questions
32.6 Further Readings

Objectives

After studying this unit, you will be able to:
Discuss the radical extension

Explain that a polynomial equation is solvable by radical

Introduction

In most results, in this section we will assume that the fields have characteristic zero, in order to
guarantee that no irreducible polynomial has multiple roots. When we say that a polynomial
equation is solvable by radicals, we mean that the solutions can be obtained from the coefficients
in a finite sequence of steps, each of which may involve addition, subtraction, multiplication,
division, or taking nth roots. Only the extraction of an nth root leads to a larger field, and so our
formal definition is phrased in terms of subfields and adjunction of roots of x -a for suitable
n
elements a.

32.1 Radical Extension

Definition: An extension field F of K is called a radical extension of K if there exist elements
u , u , ... , u in F and positive integers n , n , ... , n such that
2
m
1
2
m
1
(i) F = K (u , u , ... , u ), and
2
1
m
(ii) u is in K and u is in K ( u , ... , u ) for i = 2, ... , m .
n
n
i-1
1
1 1
i i
32.2 Solvable by Radicals
For a polynomial f(x) in K[x], the polynomial equation f(x) = 0 is said to be solvable by radicals
if there exists a radical extension F of K that contains all roots of f(x).
Proposition: Let F be the splitting field of x - 1 over a field K of characteristic zero. Then
n
Gal(F/K) is an abelian group.
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