A University Algebra: An Introduction to Classic and Modern Algebra

351 Pages·1971·18.504 MB·English

by  D. E. Littlewood

UNIVERSITY ALGEBRA
AN INTRODUCTION
TO
CLASSIC
AND
MODERN ALGEBRA
D. E. LITTLEWOOD
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A UNIVERSITY ALGEBRA

A
UNIVERSITY
ALGEBRA
AN INTRODUCTION TO CLASSIC
AND MODERN ALGEBRA
By
D. E. LITTLEWOOD
Professor of Mathematics at the University College
of North Wales, Bangor
SECOND EDITION
DOVER PUBLICATIONS, INC.
NEW YORK
This Dover edition, first published in 1970, is an
unabridged and unaltered republication of the
second (1958) edition of the work originally pub­
lished in 1950. It is reprinted by special arrangement
with Heinemann Educational Books Ltd., publisher
of the previous editions.
International Standard Book Number: 0-486-62715-2
Library of Congress Catalog Card Number: 75-139976
Manufactured in the United States of America
Dover Publications, Inc.
180 Varick Street
New York, N. Y. 10014
PREFACE
It has been my endeavour to include in this book all the Algebra
that reasonably would be required for an Honours Degree course in
Mathematics. There may be omissions but I do not think that they
can be very extensive. Some topics, especially the last chapters on
Group Representation theory, might appear to be beyond the scope
of a degree course. However, this work has such extensive applications
in other branches of mathematics as, e.g., in Quantum Theory and
Nuclear Physics, that it was felt that a concise and simplified account
could be included usefully in a non-specialist book.
Bi-altemants form a topic which may not be very familiar, but
besides the intrinsic elegance of the work it has recently acquired
new significance because of its application to Representation Theory.
The chapters are to a great extent independent of one another,
though of course some key topics are continually employed. The
teacher can thus select his material without an exhaustive study of
all the preceding chapters. Certain selected chapters, notably I, II,
V, XI, and selected parts of III, VIII and X, are recommended for a
Pass Degree course.
In the preparation and production of the book I must express my
real gratitude for very helpful cooperation to the Publishers, to their
Editor Mr. Alan Hill, and to their Reader, Professor J. L. B. Cooper.
D. E. LITTLEWOOD.
Bangor, 1950.
PREFACE TO SECOND EDITION
In addition to minor alterations and additions there is an almost
complete change in Chapter IX, and a much more detailed account
of ideals is given in Chapters IX and X. The approach to ideals is
that of classical rather than abstract algebra on the belief that a
student cannot intuitively grasp the significance of abstract methods
until he has had some manipulative experience of specific cases.
Answers to the exercises have been added. My thanks are due to
Mr. A. 0. Morris, who has calculated most of these. Sundry errors
which crept into the first edition have been eliminated.
D. E. LITTLEWOOD.
Bangor, 1958.

CONTENTS
CHAPTER PACE
I. Linear Equations and Determinants 1
Vectors—Functions of Vectors—Alternating Functions—
Alternating Linear Functions of Vectors—Determinants
—Systems of Linear Equations.
II. Matrices.........................................................................................21
Inner Products—Linear Transformations—Matrices—Trans­
position—Square Matrices—Singular and Non-singular
Matrices—Matrices and Determinants—Rank—Transform of
a Matrix—Latent Roots and Poles—Spur of a Matrix—
Applications to Coordinate Geometry.
III. Quadratic Fo r m s ...................................................................45
Reduced form—Positive Definite Forms—Law of Inertia—
Orthogonal Reduction to Reduced Form—Simultaneous
Reduction of Two Forms—Applications to Geometry.
IV. Groups 67
Definition—Classes of Conjugate Elements—Permutation
Groups—Conjugate and Self-con jugate Subgroups—Cycles—
Symmetric Group—Alternating Group—Transitive and
Intransitive Permutation Groups—Continuous Groups—
Quotient Groups.
V. Symmetric Fu n c t io n s.......................................................81
Functions Belonging to a Group—Monomial and Elementary
Symmetric Functions—Homogeneous Product Sums—Power
Sums—Relations between the Different Types of Function—
General Symmetric Function.
VI. Alternants and the General Theory of Determinants 90
Alternants—Conjugate Partitions—An Important Determi­
nant—Bi-alternants—General Theory of Determinants—
Adjugate Determinant — Laplace Development — The
$-symbols.
VII. Further Properties of Matrices . . . . 105
Characteristic Equation—Blocked Matrices—Rank of a
Product of Matrices—Canonical Form for Repeated Latent
Roots—Hermitian, Skew-Hermitian and Unitary Matrices—
Construction of Orthogonal and Unitary Matrices—Hermitian
Forms—Compound and Induced Matrices—Kronecker Pro­
duct.
VIII. Euclid’s Algorithm...................................................................127
Numerical Partial Fractions—Uniqueness of Prime Factoriza­
tion—Congruences and Residues—Congruences to Prime
Modulus—Quadratic Congruences—Theorem of Quadratic
Reciprocity—Prime Power Moduli—The General Composite
Modulus—-Simultaneous Congruences.
vll
viii CONTENTS
CHAFTEE PAGE
IX. The Laws of Al g e b r a ...................................................... 143
Laws of Arithmetic—Rings—Integral Domains—Fields—
Isomorphism—Homomorphism—Extension Rings—Ideals—
Fields of Quotients — Integers — Rationals — Indefinables—
Polynomials.
X. Polynomials............................................................................. 153
Euclid’s algorithm—Uniqueness of Factorization—Partial
Fractions—Algebraic Fields—Complex Numbers—Theory of
Algebraic Fields—Algebraic Integers—Unique Factorization
Domains—Ideals.
XI. Algebraic Equations ........................................................ 184
Eliminants—Simultaneous Equations in Two Variables—
Discriminants—Separation of Repeated Roots—Cubic and
Quartic Equations—Sturm’s Functions—Newton’s Method—
Method of Proportional Parts—Horner’s Method—Root
Squaring.
XII. Galois Theory of Equations............................................. 203
Galois Theory—Jordan-Holder Theorem—Solvability of
Symmetric Groups—Solvable Equations of Degree Greater
than 4—Rule and Compass Construction of Angles.
XIII. In v a r ia n t s.............................................................................. 221
Full Linear Group; Binary Forms—Symbolic Method—
Polarization — Seminvariants — Fundamental Theorem —
Ternary Forms—Four and More Variables—Tensors—Sum­
mation Convention — Symmetric, Skew-Symmetric and
Alternating Tensors — Variable Tensors — Fundamental
Theorem—Relation between Symbolic Method and Tensors—
Restricted Groups—Euclidean Group.
XIV. Algebras......................................................................................... 242
Definition — Order — Isomorphisms — Automorphisms —
Real Quaternions—Grassmann’s Space Algebra—Modulus—
Regular Matrix Representation—Trace—Subalgebras—Divi­
sion Algebras—Idempotents—Properly Nilpotent Elements—
Direct Products of Algebras—Fundamental Theorems.
XV. Group Algebras—The Symmetric Group . 259
Group Algebras—Matrix Representations of Groups—Group
Characters—Orthogonal Properties of Characters—Charac­
teristic Units—Symmetric Group—Schur’s Characteristic
Functions — Young’s Representation — Multiplication of
S-functions.
XVI. The Continuous Groups........................................................ 283
Full linear Group—The Reciprocal Matrix—Application to
Tensor Analysis and Invariant Theory — Plethysm of
S-functions—Orthogonal Groups—Spin Representations—
Application to Invariant Theory.
Answ ers........................................................................................ 309