Page 281 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra Richa Nandra, Lovely Professional University

Notes Unit 30: Invariant Subfield

CONTENTS
Objectives

Introduction
30.1 G-invariant Subfield
30.2 Summary
30.3 Keywords
30.4 Review Questions

30.5 Further Readings

Objectives

After studying this unit, you will be able to:
Define G-invariant subfield

Discuss examples related to subfield

Introduction

In the last unit, you have studied about computing Galois theory and groups. In this unit, you
will get information related to fundamental theorem.

30.1 G-invariant Subfield

Proposition: Let F be a field, and let G be a subgroup of Aut(F). Then

{ a in F |  (a) = a for all  in G }
is a subfield of F.
Definition: Let F be a field, and let G be a subgroup of Aut (F). Then
{ a in F |  (a) = a for all  in G }
is called the G-fixed subfield of F, or the G-invariant subfield of F, and is denoted by F .
G

Proposition: If F is the splitting field over K of a separable polynomial and G = Gal(F/K), then
F = K.
G
Lemma [Artin]: Let G be a finite group of automorphisms of the field F, and let K = F . Then
G
[F : K]  | G |.
Let F be an algebraic extension of the field K. Then F is said to be a normal extension of K if every
irreducible polynomial in K[x] that contains a root in F is a product of linear factors in F[x].

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