A Course in Algebra

[227/0%]

E. B. Vinberg

Chapter 1Algebraic Structures[14/0%]

Section 1.2Abelian Groups[1/0%]

Section 1.3Rings and Fields[5/0%]

Section 1.5Residue Rings[2/0%]

Section 1.7Algebras[2/0%]

Section 1.8Matrix Algebra[4/0%]

Chapter 2The Rudiments of Linear Algebra[16/0%]

Section 2.2Basis and Dimension of a Vector Space[9/0%]

Section 2.3Rank of a Matrix[1/0%]

Section 2.4Determinants[1/0%]

Section 2.5Some Applications of Determinants[5/0%]

Chapter 3The Rudiments of Polynomial Algebra[17/0%]

Section 3.1Construction and Basic Properties of the Polynomial Algebra[1/0%]

Section 3.2General Properties of Roots of Polynomials[1/0%]

Section 3.4Roots of Polynomials with Real Coefficients[1/0%]

Section 3.5Divisibility Theory in Euclidean Rings[5/0%]

Section 3.7Polynomials in Several Variables[2/0%]

Section 3.8Symmetric Polynomials[2/0%]

Section 3.9Cubic Equations[1/0%]

Section 3.10The Field of Rational Fractions[4/0%]

Chapter 4The Rudiments of Group Theory[22/0%]

Section 4.1Definition and Examples[4/0%]

Section 4.2Groups in Geometry and Physics[2/0%]

Section 4.3Cyclic Groups[3/0%]

Section 4.4Generating Systems[3/0%]

Section 4.5Partition into Cosets[5/0%]

Section 4.6Homomorphisms[5/0%]

Chapter 5Vector Spaces[14/0%]

Section 5.1Subspaces[3/0%]

Section 5.2Linear Maps[2/0%]

Section 5.3The Conjugate Space[3/0%]

Section 5.4Bilinear and Quadratic Functions[3/0%]

Section 5.5Euclidean Space[2/0%]

Section 5.6Hermitian Spaces[1/0%]

Chapter 6Linear Operators[11/0%]

Section 6.1The Matrix of a Linear Operator[1/0%]

Section 6.2Eigenvectors[3/0%]

Section 6.3Linear Operators and Bilinear Functions in Euclidean Space[2/0%]

Section 6.4Jordan Form[3/0%]

Section 6.5Functions of a Linear Operator[2/0%]

Chapter 7Affine and Projective Spaces[33/0%]

Section 7.1Affine Spaces[6/0%]

Section 7.2Affine Maps[6/0%]

Section 7.3Convex Sets[4/0%]

Section 7.4Euclidean Affine Spaces[7/0%]

Section 7.5Quadrics[2/0%]

Section 7.6Projective Spaces[8/0%]

Chapter 8Tensor Algebra[10/0%]

Section 8.1The Tensor Product of Vector Spaces[5/0%]

Section 8.3The Symmetric Algebra[1/0%]

Section 8.4The Grassmann Algebra[4/0%]

Chapter 9Commutative Algebra[30/0%]

Section 9.1Finitely Generated Abelian Groups[13/0%]

Section 9.2Ideals and Quotient Rings[5/0%]

Section 9.3Modules over Principal Ideal Rings[4/0%]

Section 9.4Noetherian Rings[1/0%]

Section 9.5Algebraic Extensions[3/0%]

Section 9.7Factorization into Primes[4/0%]

Chapter 10Groups[12/0%]

Section 10.1Direct and Semidirect Products[1/0%]

Section 10.2The Commutator Subgroup[1/0%]

Section 10.3Actions[3/0%]

Section 10.4Sylow's Theorems[2/0%]

Section 10.5Simple Groups[5/0%]

Chapter 11Linear Representations and Associative Algebras[28/0%]

Section 11.1Invariant Subspaces[4/0%]

Section 11.3Finite-Dimensional Associative Algebras[4/0%]

Section 11.4Linear Representations of Finite Groups[12/0%]

Section 11.5Invariants[1/0%]

Section 11.6Division Algebras[7/0%]

Chapter 12Lie Groups[20/0%]

Section 12.1Definition and Elementary Properties of Lie Groups[4/0%]

Section 12.2The Exponential Map[2/0%]

Section 12.3The Tangent Lie Algebra and the Adjoint Representation[7/0%]

Section 12.4Linear Representations of Lie Groups[7/0%]