Algebraic Theories

by Leonard Dickson

Overview

This in-depth introduction to classical topics in higher algebra provides rigorous, detailed proofs for its explorations of some of mathematics' most significant concepts, including matrices, invariants, and groups.
Algebraic Theories studies all of the important theories; its extensive offerings range from the foundations of higher algebra and the Galois theory of algebraic equations to finite linear groups (including Klein's "icosahedron" and the theory of equations of the fifth degree) and algebraic invariants. The full treatment includes matrices, linear transformations, elementary divisors and invariant factors, and quadratic, bilinear, and Hermitian forms, both singly and in pairs. The results are classical, with due attention to issues of rationality. Elementary divisors and invariant factors receive simple, natural introductions in connection with the classical form and a rational, canonical form of linear transformations. All topics are developed with a remarkable lucidity and discussed in close connection with their most frequent mathematical applications.

Product Details

ISBN-13:	9780486155203
Publisher:	Dover Publications
Publication date:	03/05/2014
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	288
File size:	12 MB
Note:	This product may take a few minutes to download.

Table of Contents

I.	Introduction to Algebraic Invariants	1
	Linear transformations
	Hessians
	Invariants and covariants
	Jacobians
	Discriminants
	Canonical forms of binary cubic and quartic forms
	Solution of cubic and quartic equations
	Homogeneity
	Weights
	Seminvariants
	Fundamental system of covariants of a binary p-ic for p [greater than sign] 5
II.	Further Theory of Covariants of Binary Forms	24
	Annihilators
	Commutators
	Existence of a covariant with a given seminvariant leader
	Hilbert's theorem
	Finiteness of a fundamental system of covariants
	Finiteness of syzygies
	Canonical form of a binary form of odd order
	List of further topics
III.	Matrices, Bilinear Forms, Linear Equations	39
	Linear forms
	Matrices
	Linear transformations
	Inverse and adjoint matrices
	Associative and distributive laws
	Characteristic equation
	Rank of a matrix
	Bilinear forms
	Fields
	Linear independence
	Equivalence of matrices or bilinear forms
	Linear equations
IV.	Quadratic and Hermitian Forms, Symmetric and Hermitian Bilinear Forms	64
	Symmetric bilinear forms and quadratic forms
	Hermitian forms and Hermitian bilinear forms
	Rational reduction of quadratic and Hermitian forms
	Canonical forms
	Reduction of a real quadratic form by real orthogonal transformation
	Rank of a symmetric or Hermitian matrix
	Kronecker's method of reduction
	Number of positive coefficients in the canonical form
V.	Theory of Linear Transformations, Invariant Factors and Elementary Divisors	89
	Rational canonical form of a linear transformation
	Invariant factors of a canonical transformation
	Rotations and orthogonal transformations
	Canonical form determined by invariant factors
	Similar transformations
	Classic canonical form
	Elementary divisors
VI.	Pairs of Bilinear, Quadratic, and Hermitian Forms	112
	Equivalence of two pairs of matrices
	Canonical forms of a pair of bilinear forms
	Pencils of bilinear forms
	The nth roots of a matrix
	Equivalence of pairs of quadratic or Hermitian forms, or symmetric or Hermitian bilinear forms
	Pairs involving alternate forms
	Existence of a pair of quadratic or Hermitian forms with any preassigned invariant factors
	Weierstrass's canonical pair of quadratic forms
	List of further applications of matrices to forms
VII.	First Principles of Groups of Substitutions	135
	Cubic and quartic equations
	Their discriminants
	Substitutions
	Groups
	Group leaving a function invariant
	Alternating group
VIII.	Fields, Reducible and Irreducible Functions	150
	Greatest common divisor
	Gauss's lemma
	Irreducibility of x[superscript p]--A and of the cyclotomic equation
IX.	Group of an Equation for a Given Field	159
	Galois resolvents
	Properties A and B of the group of an equation
	Transitive and regular groups
	Rational functions belonging to a group
	Effect on the group by an adjunction to the field
	Group of the general equation
	Outline of further results
X.	Equations Solvable by Radicals	178
	History
	Resolvent equations and their groups
	Invariant subgroup
	Transforms of a substitution
	Simple and quotient groups
	Series and factors of composition
	Solvable groups
	Equations with a cyclic group
	Cyclotomic equations
	Theorems of Jordan and Galois
	Criterion for solvability by radicals
	General equation of degree n[greater than sign]4 not solvable by radicals
	Solvable quintics
XI.	Constructions with Ruler and Compasses	204
	Analytic criterion for constructibility
	Trisection of an angle
	Regular polygons
XII.	Reduction of Equations to Normal Forms	210
	Tschirnhaus transformations
	Principal equations
	The Bring-Jerrard normal form
	Brioschi's normal form of quintic equations
XIII.	Groups of the Regular Solids; Quintic Equations	220
	Linear fractional transformation corresponding to a rotation
	Tetrahedral, octahedral, and icosahedral groups; their invariants and form problems
	Principal quintic resolvent of the icosahedral equation; its identification with any principal quintic
	General quintic
	Transformation of Brioschi's resolvent into the principal resolvent
	Galois group of the icosahedral equation
	Further results stated
XIV.	Representations of a Finite Group as a Linear Group; Group Characters	251
	Reducible linear groups
	Representations
	Irreducible and reducible group matrices
	Regular group matrix
	Group characters
	Applications to group matrices
	Alternating group on five letters
	Computation of group characters
	Subject Index	271
	Author Index	275