This is a softcover reprint of the English translation of 1990 of the revised and expanded version of Bourbaki's,
Algèbre
, Chapters 4 to 7 (1981).
This completes
Algebra
, 1 to 3, by establishing the theories of commutative fields and modules over a principal ideal domain. Chapter 4 deals with polynomials, rational fractions and power series. A section on symmetric tensors and polynomial mappings between modules, and a final one on symmetric functions, have been added. Chapter 5 was entirely rewritten. After the basic theory of extensions (prime fields, algebraic, algebraically closed, radical extension), separable algebraic extensions are investigated, giving way to a section on Galois theory. Galois theory is in turn applied to finite fields and abelian extensions. The chapter then proceeds to the study of general non-algebraic extensions which cannot usually be found in textbooks: p-bases, transcendental extensions, separability criterions, regularextensions. Chapter 6 treats ordered groups and fields and based on it is Chapter 7: modules over a p.i.d. studies of torsion modules, free modules, finite type modules, with applications to abelian groups and endomorphisms of vector spaces. Sections on semi-simple endomorphisms and Jordan decomposition have been added.
Chapter IV: Polynomials and Rational Fractions
Chapter V: Commutative Fields
Chapter VI: Ordered Groups and Fields
Chapter VII: Modules Over Principal Ideal Domains
This is a softcover reprint of the English translation of 1990 of the revised
and expanded version of Bourbaki's textbook, Algebre, Chapters 4 to 7 (1981).
The English translation of the new and expanded version of Bourbaki's Algebre,
Chapters 4 to 7 completes Algebra, 1 to 3, by establishing the theories
of commutative fields and modules over a principal ideal domain. Chapter
4 deals with polynomials, rational fractions and power series. A section
on symmetric tensors and polynomial mappings between modules, and a final
one on symmetric functions, have been added. Chapter 5 has been entirely
rewritten. After the basic theory of extensions (prime fields, algebraic,
algebraically closed, radical extension), separable algebraic extensions
are investigated, giving way to a section on Galois theory. Galois theory
is in turn applied to finite fields and abelian extensions. The chapter
then proceeds to the study of general non-algebraic extensions which cannot
usually be found in textbooks: p-bases, transcendental extensions, separability
criterions, regular extensions. Chapter 6 treats ordered groups and fields
and based on it is Chapter 7: modules over a p.i.d. studies of torsion
modules, free modules, finite type modules, with applications to abelian
groups and endomorphisms of vector spaces. Sections on semi-simple endomorphisms
and Jordan decomposition have been added. Chapter IV: Polynomials and Rational
Fractions - Chapter V: Commutative Fields - Chapter VI: Ordered Groups
and Fields - Chapter VII: Modules Over Principal Ideal Domains.
