Field theory, steven roman

Graduate Texts in Mathematics 158 Editorial Board J.H Ewing F.W Gehring P.R Halmos Graduate Texts in Mathematics 10 ll 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 TAKEUTIIZARING Introduction to Axiomatic Set Theory 2nd ed OxTOBY Measure and Category 2nd ed SCHAEFFER Topological Vector Spaces HILTON/STAMMBACH A Course in Homological Algebra MAc LANE Categories for the Working Mathematician HUGHES/PIPER Projective Planes SERRE A Course in Arithmetic TAKEUTIIZARING Axiomatic Set Theory HUMPHREYS Introduction to Lie Algebras and Representation Theory COHEN A Course in Simple Homotopy Theory CONWAY Functions of One Complex Variable 2nd ed BEALs Advanced Mathematical Analysis ANDERSON/FULLER Rings and Categories of Modules 2nd ed GoLUBITSKY/GUILEMIN Stable Mappings and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER The Siructure of Fields RosENBLATT Random Processes 2nd ed HALMOS Measure Theory HALMOS A Hilbert Space Problem Book 2nd ed HUSEMOLLER Fibre Bundles 3rd ed HUMPHREYS Linear Algebraic Groups BARNEs/MACK An Algebraic Introduction to Mathematical Logic GREUB Linear Algebra 4th ed HoLMES Geometric Functional Analysis and Its Applications HEWITT/STROMBERG Real and Abstract Analysis MANES Algebraic Theories KELLEY General Topology ZARISKIISAMUEL Commutative Algebra Vol.l ZARISKIISAMUEL Commutative Algebra Vol.ll JACOBSoN Lectures in Absiract Algebra I Basic Concepts JACOBSoN Lectures in Abstract Algebra II Linear Algebra JACOBSON Lectures in Abstract Algebra III Theory of Fields and Galois Theory 33 HIRSCH Differential Topology 34 SPITZER Principles of Random Walk 2nd ed 35 WERMER Banach Algebras and Several Complex Variables 2nd ed 36 KELLEYINAMIOKA et aJ Linear Topological Spaces 37 MONK Mathematical Logic 38 GRAUERT!FRITZSCHE Several Complex Variables 39 ARVESON An Invitation to C*-Aigebras 40 KEMENY/SNELL/KNAPP Denumerable Markov Chains 2nd ed 41 APOSTOL Modular Functions and Dirichlet Series in Number Theory 2nd ed 42 SERRE; Linear Representations of Finite Groups 43 GILLMAN/JERISON Rings of Continuous Functions 44 KENDIG Elementary Algebraic Geometry 45 LoEVE Probability Theory I 4th ed 46 LoE.vE Probability Theory II 4th ed 47 MOISE Geometric Topology in Dimensions and 48 SACHS!WU General Relativity for Mathematicians 49 GRUENBERG/WEIR Linear Geometry 2nd ed 50 EDWARDS Fermat's Last Theorem 51 KLINGENBERG A Course in Differential Geometry 52 HARTSHORNE Algebraic Geometry 53 MANIN A Course in Mathematical Logic 54 GRAVER/WATKINS Combinatorics with Emphasis on the Theory of Graphs 55 BROWN/PEARCY Introduction to Operator Theory I: Elements of Functional Analysis 56 MASSEY Algebraic Topology: An Introduction 57 CRowELL/Fox Introduction to Knot Theory 58 KoBLITZ p-adic Numbers, p-adic Analysis, and Zeta-Functions 2nd ed 59 LANG Cyclotomic Fields 60 ARNOLD Mathematical Methods in Classical Mechanics 2nd ed 61 WHITEHEAD Elements of Homotopy Theory 62 KARGAPoLOV/MERLZJAKov Fundamentals of the Theory of Groups 63 BoLLOBAS Graph Theory 64 EDWARDS Fourier Series Vol I 2nd ed continued after Index Steven Roman Field Theory Springer Science+Business Media, LLC Steven Roman Department of Mathematics California State University Fullerton, CA 92637 USA Editorial Board J.H Ewing Department of Mathematics Indiana University Bloomington, IN 47405 USA F W Gehring Department of Mathematics University of Michigan Ann Arbor, Ml 48109 USA P.R Halmos Department of Mathematics Santa Clara University Santa Clara, CA 95053 USA Mathematics Subject Classifications (1991): 12-01 With Illustrations Library of Congress Cataloging-in-Publication Data Roman, Steven Field theory I Steven Roman p em - (Graduate texts in mathematics; 158) Includes bibliographical references and indexes Algebraic fields I Title II Series QA247.R598 1995 512' 3-dc20 Galois theory Polynomials 94-36400 Printed on acid-free paper © 1995 Steven Roman Originally published by Springer-Verlag New York, Inc., in 1995 All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone Production managed by Hal Henglein; manufacturing supervised by Genieve Shaw Camera-ready copy prepared by the author using EXP® 987654321 ISBN 978-0-387-94408-1 ISBN 978-1-4612-2516-4 (eBook) DOI 10.1007/978-1-4612-2516-4 To Donna Preface This book presents the basic theory of fields, starting more or less from the beginning It is suitable for a graduate course in field theory, or independent study The reader is expected to have absorbed a serious undergraduate course in abstract algebra, not so much for the material it contains but for the oft-mentioned mathematical maturity it provides The book begins with a preliminary chapter (Chapter 0), which is designed to be quickly scanned or skipped and used as a reference if needed The remainder of the book is divided into three parts Part 1, entitled Basic Theory, begins with a chapter on polynomials Chapter is devoted to various types of field extensions In Chapter 3, we treat algebraic independence, starting with the general notion of a dependence relation and concluding with Luroth's Theorem on intermediate fields of a simple transcendental extension Chapter is devoted to the notion of separability of algebraic extensions Part of the book is entitled Galois Theory Chapter begins with the notion of a Galois correspondence between two partially ordered sets, and then specializes to the Galois correspondence of a field extension, concluding with a brief discussion of the Krull topology In Chapter 6, we discuss the Galois theory of equations In Chapter 7, we take a closer look at a finite field extension E of F as a vector space over F The next two chapters are devoted to a fairly thorough discussion of finite fields Mobius inversion is used in a few brief spots in these chapters, so an appendix has been included on this subject Part of the book is entitled The Theory of Binomials Chapter 10 covers the roots of unity (that is, the roots of the binomial xn -1) and includes Wedderburn's theorem (a finite division ring is a field) This viii Preface also seems like the appropriate time to discuss the question of whether a given group is the Galois group of a field extension In Chapter 11, we characterize the splitting fields of binomials xn- u, when the base field contains the n-th roots of unity Chapter 12 is devoted to the question of solvability of a polynomial equation by radicals (This chapter might make a convenient ending place in a graduate course.) In Chapter 13, we determine conditions that characterize the irreducibility of a binomial and describe the Galois group of a binomial Chapter 14 briefly describes the theory of families of binomials -the so-called Kummer theory Sections marked with an asterisk are optional, in that they may be skipped without loss of continuity The unmarked sections might be considered as forming a basic core course in field theory Contents Section& marked with an asterisk are optional Preface Chapter Preliminaries 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Lattices Groups Rings Integral Domains Unique Factorization Domains Principal Ideal Domains Euclidean Domains Tensor Products vii 1 12 15 17 17 18 19 Part Basic Theory 23 Chapter Polynomials 25 1.1 1.2 1.3 1.4 1.5 1.6 1.7 25 26 28 31 32 33 35 Polynomials Over a Ring Primitive Polynomials The Division Algorithm Splitting Fields The Minimal Polynomial Multiple Roots Testing for hreducibility Contents X Chapter Field Extensions 39 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 39 40 41 42 43 45 46 48 52 The Lattice of Subfields of a Field Distinguished Extensions Finitely Generated Extensions Simple Extensions Finite Extensions Algebraic Extensions Algebraic Closures Embeddings Splitting Fields and Normal Extensions Chapter Algebraic Independence 3.1 3.2 3.3 *3.4 Dependence Relations Algebraic Dependence Transcendence Bases Simple Transcendental Extensions 61 61 64 67 73 Chapter Separability 79 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 79 81 82 84 87 88 91 94 Separable Polynomials Separable Degree The Simple Case The Finite Case The Algebraic Case Pure Inseparability Separable and Purely Inseparable Closures Perfect Fields Part Galois Theory 99 Chapter Galois Theory I 101 5.1 5.2 5.3 5.4 5.5 101 104 109 112 113 Galois Connections The Galois Correspondence Who's Closed? Normal Subgroups and Normal Extensions More on Galois Groups 259 Appendix: Mobius Inversion Addition and scalar multiplication are defined on A(P) by and (f+g)(x,y) = f(x,y) + g(x,y) (kf)(x,y) = k[f(x,y)) We also define multiplication by I: (f*g)(x,y) = f(x,z)g(z,y) x~z~y the sum being finite, since P is assumed to be locally finite Using these definitions, it is not hard to show that A(P) is an algebra, called the incidence algebra of P The identity in this algebra is 6(x,y) = {! if if X= y ::j: y X The next theorem characterizes those elements of A(P) that have multiplicative inverses Theorem A.2.1 An element f E A(P) is invertible if and only if f(x,x) ::j: for all x E P Proof An inverse g of f must satisfy (A.2.1) I: f(x,z)g(z,y) = 6(x,y) x~z~y In particular, for x = y, we get f(x,x)g(x,x) = This shows the necessity and also that g(x,x) must satisfy (A.2.2) g(x,x) = f('l.x) Equation (A.2.2) defines g(x,y) when the interval [x,y] has cardinality 1, that is, when x = y We can use (A.2.1) to define g(x,y) for intervals [x,y] of all cardinalities Suppose that g(x,y) has been defined for all intervals with cardinality at most n, and let [x,y] have cardinality n+l Then, by (A.2.1), since x ::j: y, we get f(x,x)g(x,y) = - I: x x since [z,y] has cardinality at most n, and so we can use this to define g(x,y) I Definition The function ( E A.(P), defined by ((x,y) = { if X~ y if x j; y is called the zeta function Its inverse J.t(x,y) is called the Mobius function [] The next result follows from the appropriate definitions Theorem A.2.2 The Mobius function is uniquely determined by any of the following conditions 1) J.t(x,x) = and, for x < y, I: J.t(z,y) = I: J.t(x,z) = x:5z:5y 2) J.t(x,x) = and, for x < y, x:5z:5y 3) J.t(x,x) = and, for x < y, =- I: J.t{z,y) J.t(x,y) = - I: J.l(x,z) J.t(x,y) 4) J.l(x,x) x f(x) = L J.t(x,y)g(y) x:5y Proof Since all sums are finite, we have, for any x, 261 Appendix: Mobius Inversion y~x g(y)f.l(y,x) = y~x [z~y f(z)f(y,x) = L L z$;x z$;y$;x f(z)Jl(y,x) I: f{z) I: Jl(y,x) z$y$;x = L f(z)c5(z,x) = f(x) = z