Algebra II: Chapters 4–7

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ELEMENTS OF MATHEMATICS

Algebra II: Chapters 4-7

Copyright © Masson, 1990

     ISBN 3-540-19375-

     ISBN 0-387-19375-8

     ISBN 0-387-19375-8 (Vol. 2: U.S.).

     0A155. B68213 1988 512 88-31211.

To the reader

CHAPTER IV  Polynomials and rational fractions

     § 1. POLYNOMIALS

          1. Definition of polynomials

          2. Degrees

          3. Substitutions

          4. Differentials and derivations

          5. Divisors of zero in a polynomial ring

          6. Euclidean division of polynomials in one indeterminate

          7. Divisibility of polynomials in one indeterminate

          8. Irreducible polynomials

     § 2. ZEROS OF POLYNOMIALS

          1. Roots of a polynomial in one indeterminate. Multiplicity

          2. Differential criterion for the multiplicity of a root

          3. Polynomial functions on an infinite integral domain

     § 3. RATIONAL FRACTIONS

          1. Definition of rational fractions

          2. Degrees

          3. Substitutions

          4. Differentials and derivations

     § 4. FORMAL POWER SERIES

          1. Definition of formal power series. Order

          2. Topology on the set of formal power series. Summable families

          3. Substitutions

          4. Invertible formal power series

          5. Taylor\'s formula for formal power series

          6. Derivations in the algebra of formal power series

          7. The solution of equations in a formal power series ring

          8. Formal power series over an integral domain

          9. The field of fractions of the ring of formal power series in one indeterminate over a field

          10. Exponential and logarithm

     § 5. SYMMETRIC TENSORS AND POLYNOMIAL MAPPINGS

          1. Relative traces

          2. Definition of symmetric tensors

          3. Product for symmetric tensors

          4. Divided powers

          5. Symmetric tensors over a free module

          6. The functor TS

          7. Coproduct for symmetric tensors

          8. Relations between T S (M) and S (M)

          9. Homogeneous polynomial mappings

          10. Polynomial mappings

          11. Relations between S (M * ), T S (M) *gr and Pol (M, A)

     § 6. SYMMETRIC FUNCTIONS

          1. Symmetric polynomials

          2. Symmetric rational fractions

          3. Symmetric formal power series

          4. Sums of powers

          5. Symmetric functions in the roots of a polynomial

          6. The resultant

          7. The discriminant

     Exercises

CHAPTER V  Commutative Fields

     § 1. PRIME FIELDS. CHARACTERISTIC

          1. Prime fields

          2. Characteristic of a ring and of a field

          3. Commutative rings of characteristic p

          4. Perfect rings of characteristic p

          5. Characteristic exponent of a field. Perfect fields

          6. Characterization of polynomials with zero differential

     § 2. EXTENSIONS

          1. The structure of an extension

          2. Degree of an extension

          3. Adjunction

          4. Composite extensions

          5. Linearly disjoint extensions

     § 3. ALGEBRAIC EXTENSIONS

          1. Algebraic elements of an algebra

          2. Algebraic extensions

          3. Transitivity of algebraic extensions. Fields that are relatively algebraically closed in an extension field

     § 4. ALGEBRAICALLY CLOSED EXTENSIONS

          1. Algebraically closed fields

          2. Splitting extensions

          3. Algebraic closure of a field

     § 5. p -RADICAL EXTENSIONS

          1. pradical elements

          2. pradical extensions

     § 6. ETALE ALGEBRAS

          1. Linear independence of homomorphisms

          2. Algebraic independence of homomorphisms

          3. Diagonalizable algebras and etale algebras

          4. Subalgebras of an etale algebra

          5. Separable degree of a commutative algebra

          6. Differential characterization of etale algebras

          7. Reduced algebras and etale algebras

     § 7. SEPARABLE ALGEBRAIC EXTENSIONS

          1. Separable algebraic extensions

          2. Separable polynomials

          3. Separable algebraic elements

          4. The theorem of the primitive element

          5. Stability properties of separable algebraic extensions

          6. A separability criterion

          7. The relative separable algebraic closure

          8. The separable closure of a field

          9. Separable and inseparable degrees of an extension of finite degree

     § 8. NORMS AND TRACES

          1. Recall

          2. Norms and traces in etale algebras

          3. Norms and traces in extensions of finite degree

     § 9. CONJUGATE ELEMENTS AND QUASI-GALOIS EXTENSIONS

          1. Extension of isomorphisms

          2. Conjugate extensions. Conjugate elements

          3. Quasi-Galois extensions

          4. The quasi-Galois extension generated by a set

     § 10. GALOIS EXTENSIONS

          1. Definition of Galois extensions

          2. The Galois group

          3. Topology of the Galois group

          4. Galois descent

          5. Galois cohomology

          6. Artin\'s theorem

          7. The fundamental theorem of Galois theory

          8. Change of base field

          9. The normal basis theorem

          10. Finite IF-sets and etale algebras

          11. The structure of quasi Galois extensions

     § 11. ABELIAN EXTENSIONS

          1. Abelian extensions and the abelian closure

          2. Roots of unity

          3. Primitive roots of unity

          4. Cyclotomic extensions

          5. Irreducibility of cyclotomic polynomials

          6. Cyclic extensions

          7. Duality of Z/nZ-modules

          8. Kummer theory

          9. Artin-Schreier theory

     § 12. FINITE FIELDS

          1, The structure of finite fields

          2. Algebraic extensions of a finite field

          3. The Galois group of the algebraic closure of a finite field

          4. Cyclotomic polynomials over a finite field

     § 13. p-RADICAL EXTENSIONS OF HEIGHT ` 3

          1. p-free subsets and p ba s e s

          2. Differentials and pbases

          3. The correspondence between subfields and Lie algebras of derivations

     § 14. TRANSCENDENTAL EXTENSIONS

          1. Algebraically free families. Pure extensions

          2. Transcendence bases

          3. The transcendence degree of an extension

          4. Extension of isomorphisms

          5. Algebraically disjoint extensions

          6. Algebraically free families of extensions

          7. Finitely generated extensions

     § 15. SEPARABLE EXTENSIONS

          1. Characterization of the nilpotent elements of a ring

          2. Separable algebras

          3. Separable extensions

          4. Mac Lane\'s separability criterion

          5. Extensions of a perfect field

          6. The characterization of separability by automorphisms

     § 16. DIFFERENTIAL CRITERIA OF SEPARABILITY

          1. Extension of derivations .the case of rings

          2. Extension of derivations : the case of fields

          3. Derivations in fields of characteristic zero

          4. Derivations in separable extensions

          5. The index of a linear mapping

          6. Differential properties of finitely generated extensions

          7. Separating transcendence bases

     § 17. REGULAR EXTENSIONS

          1. Complements on the relative separable algebraic closure

          2. The tensor product of extensions

          3. Regular algebras

          4. Regular extensions

          5. Characterization of regular extensions

          6. Application to composite extensions

     Exercises

Historical Note

     (Chapters IV and V)

Bibliography

CHAPTER VI  Ordered groups and fields

     § 1. ORDERED GROUPS. DIVISIBILITY

          1. Definition of ordered monoids and groups

          2. Pre-ordered monoids and groups

          3. Positive elements

          4. Filtered groups

          5. Divisibility relations in a field

          6. Elementary operations on ordered groups

          7. Increasing homomorphisms of ordered groups

          8. Suprema and infirm in an ordered group

          9. Lattice ordered groups

          10. The decomposition theorem

          11. Positive and negative parts

          12. Coprime elements

          13. Irreducible elements

     § 2. ORDERED FIELDS

          1. Ordered rings

          2. Ordered fields

          3. Extensions of ordered fields

          4. Algebraic extensions of ordered fields

          5. Maximal ordered fields

          6. Characterisation of maximal ordered fields. Euler-Lagrange Theorem

          7. Vector spaces over an ordered field

     Exercises

CHAPTER VII  Modules over principal ideal domains

     § 1. PRINCIPAL IDEAL DOMAINS

          1. Definition of a principal ideal domain

          2. Divisibility in principal ideal domains

          3. Decomposition into irreducible factors in principal ideal domains

          4. Divisibility of rational integers

          5. Divisibility of polynomials in one indeterminate over a field

     § 2. TORSION MODULES OVER A PRINCIPAL IDEAL DOMAIN

          1. Modules over a product of rings

          2. Canonical decomposition of a torsion module over a principal ideal domain

          3. Applications : I. Canonical decompositions of rational numbers and of rational functions in one indeterminate

          4. Applications : II. The multiplicative group of units of the integers modulo a

     § 3. FREE MODULES OVER A PRINCIPAL IDEAL DOMAIN

     § 4. FINITELY GENERATED MODULES OVER A PRINCIPAL IDEAL DOMAIN

          1. Direct sums of cyclic modules

          2. Content of an element of a free module

          3. Invariant factors of a submodule

          4. Structure of finitely generated modules

          5. Calculation of invariant factors

          6. Linear mappings of free modules, and matrices over a principal ideal domain

          7. Finitely generated abelian groups

          8. Indecomposable modules. Elementary divisors

          9. Duality in modules of finite length over a principal ideal domain

     § 5. ENDOMORPHISMS OF VECTOR SPACES

          1. The module associated to an endomorphism

          2. Eigenvalues and eigenvectors

          3. Similarity invariants of an endomorphism

          4. Triangularisable endomorphisms

          5. Properties of the characteristic polynomial : trace and determinant

          6. Characteristic polynomial of the tensor product of two endomorphisms

          7. Diagonalisable endomorphisms

          8. Semi-simple and absolutely semi-simple endomorphisms

          9. Jordan decomposition

     Exercises

Historical Note

     (Chapters VI and VII)

Bibliography

Index of notation

Index of terminology

Table of contents