Basic Algebra I

ISBN-10: 0486471896

ISBN-13: 9780486471891

Edition: 2nd 2009

Authors: Nathan Jacobson
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Description: Volume I of a pair of classic texts and standard references for a generation this book is the work of an expert algebraist who taught at Yale for two decades. Volume I covers all undergraduate topics, including groups, rings, modules, Galois theory, polynomials, linear algebra, and associative algebra. 1974 edition.

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Book details

List price: $31.00
Edition: 2nd
Copyright year: 2009
Publisher: Dover Publications, Incorporated
Publication date: 6/22/2009
Binding: Paperback
Pages: 528
Size: 6.25" wide x 9.25" long x 1.00" tall
Weight: 1.716
Language: English

Preface to the First Edition
Introduction: Concepts from set Theory. The Integers
The power set of a set
The Cartesian product set. Maps
Equivalence relations. Factoring a map through an equivalence relation
The natural numbers
The number system Z of integers
Some basic arithmetic facts about Z
A word on cardinal numbers
Monoids and Groups
Monoids of transformations and abstract monoids
Groups of transformations and abstract groups
Isomorphism. Cayley's theorem
Generalized associativity. Commutativity
Submonoids and subgroups generated by a subset. Cyclic groups
Cycle decomposition of permutations
Orbits. Cosets of a subgroup
Congruences. Quotient monoids and groups
Subgroups of a homomorphic image. Two basic isomorphism theorems
Free objects. Generators and relations
Groups acting on sets
Sylow's theorems
Definition and elementary properties
Types of rings
Matrix rings
Ideals, quotient rings
Ideals and quotient rings for Z
Homomorphisms of rings. Basic theorems
Field of fractions of a commutative domain
Polynomial rings
Some properties of polynomial rings and applications
Polynomial functions
Symmetric polynomials
Factorial monoids and rings
Principal ideal domains and Euclidean domains
Polynomial extensions of factorial domains
"Rngs" (rings without unit)
Modules over a Principal Ideal Domain
Ring of endomorphisms of an abelian group
Left and right modules
Fundamental concepts and results
Free modules and matrices
Direct sums of modules
Finitely generated modules over a p.i.d. Preliminary results
Equivalence of matrices with entries in a p.i.d
Structure theorem for finitely generated modules over a p.i.d
Torsion modules, primary components, invariance theorem
Applications to abelian groups and to linear transformations
The ring of endomorphisms of a finitely generated module over a p.i.d
Galois Theory of Equations
Preliminary results, some old, some new
Construction with straight-edge and compass
Splitting field of a polynomial
Multiple roots
The Galois group. The fundamental Galois pairing
Some results on finite groups
Galois' criterion for solvability by radicals
The Galois group as permutation group of the roots
The general equation of the nth degree
Equations with rational coefficients and symmetric group as Galois group
Constructible regular n-gons
Transcendence of e and p. The Lindemann-Weierstrass theorem
Finite fields
Special bases for finite dimensional extensions fields
Traces and norms
Mod p reduction
Real Polynomial Equations and Inequalities
Ordered fields. Real closed fields
Sturm's theorem
Formalized Euclidean algorithm and Sturm's theorem
Elimination procedures. Resultants
Decision method for an algebraic curve
Tarski's theorem
Metric Vector Spaces and the Classical Groups
Linear functions and bilinear forms
Alternate forms
Quadratic forms and symmetric bilinear forms
Basic concepts of orthogonal geometry
Witt's cancellation theorem
The theorem of Cartan-Dieudonne
Structure of the general linear group GLn(F)
Structure of orthogonal groups
Symplectic geometry. The symplectic group
Orders of orthogonal and symplectic groups over a finite field
Postscript on hermitian forms and unitary geometry
Algebras over a Field
Definition and examples of associative algebras
Exterior algebras. Application to determinants
Regular matrix representations of associative algebras. Norms and traces
Change of base field. Transitivity of trace and norm
Non-associative algebras. Lie and Jordan algebras
Hurwitz' problem. Composition algebras
Frobenius' and Wedderburn's theorems on associative division algebras
Lattices and Boolean Algebras
Partially ordered sets and lattices
Distributivity and modularity
The theorem of Jordan-Holder-Dedekind
The lattice of subspaces of a vector space. Fundamental theorem of projective geometry
Boolean algebras
The Mobius function of a partially ordered set
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