AMS Chelsea Publishing 1975; 4022 pp; hardcover Volume: 273 ISBN10: 0828402736 ISBN13: 9780828402736 List Price: US$158 Member Price: US$142.20 Order Code: CHEL/273
 The set of Collected Mathematical Papers of Leonard Eugene Dickson contains nearly all of his research papers and two of his books, the second being the famous Algebren und ihre Zahlentheorie. The work includes an annotated Bibliography and Index. Table of Contents Part 1  Determination of the structure of all linear homogeneous groups in a Galois field which are defined by a quadratic invariant
 Certain subgroups of the BettiMathieu group
 Canonical form of a linear homogeneous substitution in a Galois field
 Subgroups of order a power of \(p\) in the general and special \(m\)ary linear homogeneous groups in the \(GF[p^n]\)
 Distribution of the ternary linear homogeneous substitutions in a Galois field into complete sets of conjugate substitutions
 Representation of linear groups as transitive substitution groups
 Analytic functions suitable to represent substitutions
 Cyclic subgroups of the simple ternary linear fractional group in a Galois field
 Canonical form of a linear homogeneous transformation in an arbitrary realm of rationality
 Determination of the ternary modular groups
 On the quaternary linear homogeneous groups modulo \(p\) of order a multiple of \(p\)
 On the canonical forms and automorphs of ternary cubic forms
 Rational reduction of a pair of binary quadratic forms; their modular invariants
 A theory of invariants
 Binary modular groups and their invariants
 Concerning the cyclic subgroups of the simple group \(G\) of all linear fractional substitutions of determinant unity in two nonhomogeneous variables with coefficients in an arbitrary Galois field
 Invariantive reduction of quadratic forms in the \(GF[2^n]\)
 Finiteness of the odd perfect and primitive abundant numbers with \(n\) distinct prime factors
 Even abundant numbers
 Invariantive theory of plane cubic curves modulo 2
 Determination of all groups of binary linear substitutions with integral coefficients taken modulo 3 and of determinant unity
 On the real elements of certain classes of geometrical configurations
 On the rank of a symmetrical matrix
 Ternary orthogonal group in a general field
 Groups defined for a general field by the rotation groups
 Lower limit for the number of sets of solutions of \(x^e +y^e +z^e \equiv 0\, (\text{mod } p)\)
 On the congruence \(x^n +y^n +z^n \equiv 0\, (\text{mod } p)\)
 Universal Waring theorems with cubic summands
 Universal forms \(a_{i}x^{n}_{i}\) and Waring's problem
 Quaternary quadratic forms representing all integers
 Sur plusieurs groupes linéaires isomorphes au groupe simple d'ordre 25920
 Sur une généralisation du théorème de Fermat
 Some fallacies of an angle trisector
 A new solution of the cubic equation
 On systems of isothermal curves
 A matrix defined by the quaternion group
 A generalization of symmetric and skewsymmetric determinants
 A property of the group \(G_{2}^{2n}\) all of whose operators except identity are of period 2
 On the factorization of large numbers
 On the representation of numbers as the sum of two squares [with M. Kaba]
 Amicable number triples
 Rational edged cuboids with equal volumes and equal surfaces
 Extensions of Waring's theorem on nine cubes
 A new method for universal Waring theorems with details for seventh powers
 Quadratic functions or forms, sums of whose values give all positive integers
 Projective classification of cubic surfaces, modulo 2
 The invariants, seminvariants and linear covariants of the binary quartic form modulo 2
 Fermat's last theorem and the origin and nature of the theory of algebraic numbers
 On quaternions and their generalization and the history of the eight square theorem
 Ternary quadratic forms and congruences
 A new definition of the general Abelian linear group
 Determination of an abstract simple group of order \(2^7 \cdot 3^6 \cdot 5\cdot 7\) holoedrically isomorphic with a certain orthogonal group and with a certain hyperabelian group
 Index
Part 2  Canonical forms of quaternary Abelian substitutions in an arbitrary Galois field
 Theory of linear groups in an arbitrary field
 On the group defined for any given field by the multiplication table of any given finite group
 The groups of Steiner in problems of contact
 On the reducibility of linear groups
 The groups of Steiner in problems of contact (second paper)
 Definitions of a field by independent postulates
 Definitions of a linear associative algebra by independent postulates
 On the subgroups of order a power of \(p\) in the quaternary Abelian group in the Galois field of order \(p^n\)
 The subgroups of order a power of 2 of the simple quinary orthogonal group in the Galois field of order \(p^n =8l\pm 3\)
 On hypercomplex number systems
 The minimum degree \(\tau\) of resolvents for the \(p\)section of the periods of hyperelliptic functions of four periods
 On commutative linear algebras in which division is always uniquely possible
 Linear algebras in which division is always uniquely possible
 Invariants of binary forms under modular transformations
 Modular theory of groupmatrices
 Representations of the general symmetric group as linear groups in finite and infinite fields
 Definite forms in a finite field
 General theory of modular invariants
 Linear algebras
 Proof of the finiteness of modular covariants
 Linear associative algebras and Abelian equations
 Quartic curves modulo 2
 Determination of all general homogeneous polynomials expressible as determinants with linear elements
 New division algebras
 Singular case of pairs of bilinear, quadratic, or Hermitian forms
 Universal quadratic forms
 Simpler proofs of Waring's theorem on cubes, with various generalizations
 Construction of division algebras
 Waring's problem for cubic functions
 A new method for Waring theorems with polynomial summands
 Cyclotomy when \(e\) is composite
 Cyclotomy and trinomial congruences
 A new method for Waring theorems with polynomial summands, II
 Some relations between the theory of numbers and other branches of mathematics
 Homogeneous polynomials with a multiplication theorem
 Linear algebras with associativity not assumed
 The ideal Waring theorem for twelfth powers
 On the inscription of regular polygons
 The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group
 A generalization of Fermat's theorem
 An elementary exposition of Frobenius's theory of groupcharacters and groupdeterminants
 Index
Part 3  Differential equations from the group standpoint
 On Waring's problem and its generalization
 The known systems of simple groups and their interisomorphisms
 The simplest model for illustrating the conic sections
 On the trisection of an angle and the construction of regular polygons of 7 and 9 sides
 Notes on the theory of numbers
 On the cyclotomic function
 Problems
 Generalizations of Waring's theorem on fourth, sixth, and eighth powers
 Additive number theory for all quadratic functions
 Minimum decompositions into \(n\)th powers
 Twofold generalizations of Cauchy's lemma
 Cyclotomy, higher congruences, and Waring's problem
 Cyclotomy, higher congruences, and Waring's problem, II
 Waring theorems of new type
 Proof of the ideal Waring theorem for exponents 7180
 Solution of Waring's problem
 Universal Waring theorems
 The structure of the linear homogeneous groups defined by the invariant
 The alternating group on eight letters and the quaternary linear congruence group modulo two
 The hyperorthogonal groups
 Universal Waring theorem for eleventh powers
 Arithmetic of quaternions
 The rational linear algebras of maximum and minimum ranks
 The straight lines on modular cubic surfaces
 Recent progress in the theories of modular and formal invariants and in modular geometry
 Quaternions and their generalizations
 Quadratic forms which represent all integers
 Polygonal numbers and related Waring problems
 New Waring theorems for polygonal numbers
 Congruences involving only \(e\)th powers
 Outline of the theory to date of the arithmetics of algebras
 Further development of the theory of arithmetics of algebras
 A new theory of linear transformations and pairs of bilinear forms
 On the minimum degree of resolvents for the \(p\)section of the periods of hyperelliptic functions of four periods
 On the theory of numbers and generalized quaternions
 Algebraic theory of the expressibility of cubic forms as determinants, with application to Diophantine analysis
 On finite algebras
 Index
Part 4  Algebren und ihre Zahlentheorie
 The structure of certain linear groups with quadratic invariants
 The group of linear homogeneous substitutions on \(mq\) variables which is defined by the invariant
 Concerning the four known simple linear groups of order 25920, with an introduction to the hyperabelian linear groups
 The abstract group isomorphic with the symmetric group on \(k\) letters
 An abstract simple group of order 25920
 Linear substitutions commutative with a given substitution
 Concerning the Abelian and related linear groups
 Generational relations for the abstract group simply isomorphic with the linear fractional group in the \(GF[2^n]\)
 Linear groups in an infinite field
 A quadratic Cremona transformation defined by a conic
 Two systems of subgroups of the quaternary Abelian group in a general Galois field
 On the subgroups of order a power of \(p\) in the linear homogeneous and fractional groups in the \(GF[p^n]\)
 On the class of the substitutions of various linear groups
 The group of a tactical configuration
 A general theorem on algebraic numbers
 Proof of the existence of the Galois field of order \(p^r\) for every integer \(r\) and prime number \(p\)
 The symmetric group on eight letters and the senary first hypoabelian group
 On quadratic forms in a general field
 On triple algebras and ternary cubic forms
 Criteria for the irreducibility of a reciprocal equation
 Modular theory of group characters
 On higher congruences and modular invariants
 On the factorization of integral functions with \(p\)adic coefficients
 On the representation of numbers by modular forms
 On the negative discriminants for which there is a single class of positive primitive binary quadratic forms
 Invariants, seminvariants, and covariants of the ternary and quaternary quadratic form modulo 2
 On the relation between linear algebras and continuous groups
 Mathematics in war perspective. [Presidential address delivered before the American Mathematical Society, December 27, 1918.]
 Fallacies and misconceptions in Diophantine analysis
 A new method in Diophantine analysis
 Index
Part 5  Recent progress on Waring's theorem and its generalizations
 Proof of a Waring theorem on fifth powers
 Waring's problem for ninth powers
 The converse of Waring's problem
 The Waring problem and its generalizations
 A generalization of Waring's problem
 All integers except 23 and 239 are sums of eight cubes
 The theory of numbers: its principal branches
 Recent light on classic problems in the theory of numbers
 The quadratic Cremona transformation
 Systems of simple groups derived from the orthogonal group
 Systems of simple groups derived from the orthogonal group
 Why it is impossible to trisect an angle or to construct a regular polygon of seven or nine sides by ruler and compasses
 Systems of continuous and discontinuous simple groups
 Higher irreducible congruences
 The structure of the hypoabelian groups
 Report on the recent progress in the theory of linear groups
 Orthogonal group in a Galois field
 Concerning a linear homogeneous group in \(C_{m, q}\) variables isomorphic to the general linear homogeneous group in \(m\) variables
 Systems of simple groups derived from the orthogonal group
 Concerning real and complex continuous groups
 The configurations of the 27 lines on a cubic surface and the 28 bitangents to a quartic curve
 An extension of the theory of numbers by means of correspondences between fields
 Integral solutions of \(x^2my^2=zw\)
 All integral solutions of \(ax^2 + bxy + cy^2 =w_{1}w_{2}\cdots w_{n}\)
 A generalization of Waring's theorem on nine cubes
 Integers represented by positive ternary quadratic forms
 Extensions of Waring's theorem on fourth powers
 All positive integers are sums of values of a quadratic function of \(x\)
 Generalizations of the theorem of Fermat and Cauchy on polygonal numbers
 Extended polygonal numbers
 New division algebras
 The forms \(ax^2 + by^2 + cz^2\) which represent all integers
 Perfect and amicable numbers
 Cyclic numbers
 Proof of the nonisomorphism of the simple Abelian group on \(2m\) indices and the simple orthogonal group on \(2m +1\) indices for \(m>2\)
 A triply infinite system of simple groups
 The first hypoabelian group generalized
 A class of groups in an arbitrary realm connected with the configuration of the 27 lines on a cubic surface
 Theorems on the residues of multinomial coefficients with respect to a prime modulus
 Determination of all the subgroups of the three highest powers of \(p\) in the group \(G\) of all \(m\)ary linear homogeneous transformations modulo \(p\)
 On the last theorem of Fermat
 On the last theorem of Fermat (second paper)
 On families of quadratic forms in a general field
 On nonvanishing forms
 Theorems and tables on the sum of the divisors of a number
 On commutative linear groups
 Modular invariants of the system of a binary cubic, quadratic, and linear form
 Gergonne's pile problem
 Isomorphism between certain systems of simple linear groups
 Impossibility of restoring unique factorization in a hypercomplex arithmetic
 Algebras and their arithmetics
 Quadratic fields in which factorization is always unique
 Resolvent sextics of quintic equations
 Obituary: Hans Frederik Blichfeldt 18731945
 Bibliography
 Index
Part 6  Algebraic invariants
 Simplicity of the Abelian group on two pairs of indices in the Galois field of order \(2^{n}, n>1\)
 A class of linear groups including the Abelian group
 The abstract form of the special linear homogeneous group in an arbitrary field
 The abstract form of the Abelian linear groups
 A class of groups in an arbitrary realm connected with the configuration of the 27 lines on a cubic surface [Second paper]
 Note on modular invariants
 Les polynomes égaux à des déterminants
 La composition des polynomes
 The largest linear homogeneous groups with an invariant Pfaffian
 The known finite simple groups
 On the groups defined for an arbitrary field by the multiplication tables of certain finite groups
 The abstract group simply isomorphic with the group of linear fractional transformations in a Galois field
 Generational relations of an abstract simple group of order 4080
 Invariants of the general quadratic form modulo 2
 Modular invariants of a general system of linear forms
 Invariantive classification of pairs of conics modulo 2
 Definition of the Abelian, the two hypoabelian, and related linear groups of the groups of isomorphism of certain elementary groups
 Determination of all the subgroups of the known simple group of order 25920
 Definition of a group and a field by independent postulates
 On semigroups and the general isomorphism between infinite groups
 On quadratic, Hermitian and bilinear forms
 Equivalence of pairs of bilinear or quadratic forms under rational transformation
 An invariantive investigation of irreducible binary modular forms
 A fundamental system of invariants of the general modular linear group with a solution of the form problem
 Invariants in the theory of numbers
 On the number of inscriptible regular polygons
 Three sets of generational relations defining the abstract simple group of order 504
 Generational relations defining the abstract simple group, of order 660
 The abstract group \(G\) simply isomorphic with the alternating group on six letters
 Criteria for the irreducibility of functions in a finite field
 On the theory of equations in a modular field
 On binary modular groups and their invariants
 Geometrical and invariantive theory of quartic curves modulo 2
 A quadratic Cremona transformation defined by a conic
 Factors of a certain determinant of order six
 The order of a certain senary linear group
 Three algebraic notes
 Note on the volume of a tetrahedron in terms of the coordinates of the vertices
 Rational triangles and quadrilaterals
 The points of inflexion of a plane cubic curve
 Reducible cubic forms expressible rationally as determinants
 A fundamental system of covariants of the ternary cubic form
 A new simple theory of hypercomplex integers
 Algèbres nouvelles de division
 Memoir on Abelian transformations
 Application of groups to a complex problem in arrangements
 Note on cubic equations and congruences
 A class of simply transitive linear groups
 Fields whose elements are linear differential expressions
 Bibliography
 Index
