Introduction to Algebra

I. Foundations of Algebra.- Further reading.- 1. Sources of algebra.- §1. Algebra in brief.- §2. Some model problems.- 1. Solvability of equations in radicals.- 2. The states of a molecule.- 3. Coding information.- 4. The heated plate problem.- §3. Systems of linear equations. The first steps.- 1. Terminology.- 2. Equivalence of linear systems.- 3. Reducing to step form.- 4. Studying a system of linear equations.- 5. Some remarks and examples.- §4. Determinants of small order.- Exercises.- §5. Sets and mappings.- 1. Sets.- 2. Mappings.- Exercises.- §6. Equivalence relations. Quotient maps.- 1. Binary relations.- 2. Equivalence relations.- 3. Quotient maps.- 4. Ordered sets.- Exercises.- §7. The principle of mathematical induction.- §8. Integer arithmetic.- 1. The fundamental theorem of arithmetic.- 2. g.c.d. and l.c.m. in ZZ.- 3. The division algorithm in ZZ.- Exercises.- 2. Vector spaces. Matrices.- §1. Vector spaces.- 1. Motivation.- 2. Basic definitions.- 3. Linear combinations. Linear span.- 4. Linear dependence.- 5. Bases. Dimension.- Exercises.- §2. The rank of a matrix.- 1. Back to equations.- 2. The rank of a matrix.- 3. Solvability criterion.- Exercises.- §3. Linear maps. Matrix operations.- 1. Matrices and maps.- 2. Matrix multiplication.- 3. Square matrices.- Exercises.- §4. The space of solutions.- 1. Solving a homogeneous linear system.- 2. Linear manifolds. Solving a non-homogeneous system.- 3. The rank of a product of matrices.- 4. Equivalence classes of matrices.- Exercises.- 3. Determinants.- §1. Determinants: construction and basic properties.- 1. Construction by induction.- 2. Basic properties of determinants.- Exercises.- §2. Further properties of determinants.- 1. Expanding the determinant along an arbitrary column.- 2. The properties of determinants relating to columns.- 3. The transpose determinant.- 4. Determinants of special matrices.- 5. Building up a theory of determinants.- Exercises.- §3. Applications of determinants.- 1. Criterion for a matrix to be non-singular.- 2. Computing the rank of a matrix.- Exercises.- 4. Algebraic structures (groups, rings, fields).- §1. Sets with algebraic operations.- 1. Binary operations.- 2. Semigroups and monoids.- 3. Generalized associativity; powers.- 4. Invertible elements.- Exercises.- §2. Groups.- 1. Definition and examples.- 2. Systems of generators.- 3. Cyclic groups.- 4. The symmetric group and the alternating group 153 Exercises.- §3. Morphisms of groups.- 1. Isomorphisms.- 2. Komomorphisms.- 3. Glossary. Examples.- 4. Cosets of a subgroup.- 5. The monomorphism Sn ? GN(n).- Exercises.- §4. Rings and fields.- 1. The definition and general properties of rings.- 2. Congruences. The ring of residue classes.- 3. Ring homomorphisms and ideals.- 4. The concept of quotient group and quotient ring.- 5. Types of rings. Fields.- 6. The characteristic of a field.- 7. A remark on linear systems.- Exercises.- 5. Complex numbers and polynomials.- §1. The field of complex numbers.- 1. An auxiliary construction.- 2. The complex plane.- 3. Geometrical interpretation of operations with complex numbers.- 4. Raising to powers and extracting roots.- 5. Uniqueness theorem.- Exercises.- §2. Rings of polynomials.- 1. Polynomials in one variable.- 2. Polynomials in several variables.- 3. The division algorithm.- Exercises.- §3. Factoring in polynomial rings.- 1. Elementary divisibility properties.- 2. g.c.d. and l.c.m. in rings.- 3. Unique factorization in Euclidean rings.- 4. Irreducible polynomials.- Exercises.- §4. The field of fractions.- 1. Construction of the field of fractions of an integral domain.- 2. The field of rational functions.- 3. Primary rational functions.- Exercises.- 6. Roots of polynomials.- §1. General properties of roots.- 1. Roots and linear factors.- 2. Polynomial functions.- 3. Differentiation in polynomial rings.- 4. Multiple factors.- 5. Vieta’s formulas.- Exercises.- §2. Symmetric polynomials.- 1. The ring of symmetric polynomials.- 2. The fundamental theorem on symmetric polynomials.- 3. The method of undetermined coefficients.- 4. The discriminant of a polynomial.- 5. The resultant.- Exercises.- §3. (E is algebraically closed.- 1. Statement of the fundamental theorem.- 2. The splitting field of a polynomial.- 3. Proof of the Fundamental Theorem.- §4. Polynomials with real coefficients.- 1. Factorization in IR[X].- 2. The problem of isolating the roots of a polynomial.- 3. Stable polynomials.- Exercises.- II. Groups, Rings, Modules.- Further reading.- 7. Groups.- §1. Classical groups in low dimensions.- 1. General definitions.- 2. Parametrization of SU(2) and SO(3).- 3. The epimorphism SU(2) ? SO(3).- 4. Geometrical characterization of SO(3).- Exercises.- §2. Group actions on sets.- 1. Homomorphisms G ? S(ft).- 2. The orbit and stationary subgroup of a point.- 3. Examples of group actions on sets.- 4. Homogeneous spaces.- Exercises.- §3. Some group theoretic constructions.- 1. General theorems on group homomorphisms.- 2. Solvable groups.- 3. Simple groups.- 4. Products of groups.- 5. Generators and defining relations.- Exercises.- §4. The Sylow theorems.- Exercises.- §5. Finite abelian groups.- 1. Primary abelian groups.- 2. The structure theorem for finite abelian groups 381 Exercises.- 8. Elements of representation theory.- §1. Definitions and examples of linear representations.- 1. Basic concepts.- 2. Examples of linear representations.- Exercises.- §2. Unitary and reducible representations.- 1. Unitary representations.- 2. Complete reducibility.- Exercises.- §3. Finite rotation groups.- 1. The orders of finite subgroups of SO(3).- 2. Symmetry groups for regular polyhedra.- Exercises.- §4. Characters of linear representations.- 1. Schur’s lemma and corollary.- 2. Characters of representations.- Exercises.- §5. Irreducible representations of finite groups.- 1. The number of irreducible representations.- 2. The degrees of the irreducible representations.- 3. Representations of abelian groups.- 4. Representations of certain special groups.- Exercises.- §6. Representations of SU(2) and SO(3).- Exercises.- §7. Tensor products of representations.- 1. The dual representation.- 2. Tensor products of representations.- 3. The ring of characters.- 4. Invariants of linear groups.- Exercises.- 9. Toward a theory of fields, rings and modules.- §1. Finite field extensions.- 1. Primitive elements and the degree of an extension.- 2. Isomorphism of splitting fields.- 3. Finite fields.- 4. The Mobius inversion formula and its applications.- Exercises.- §2. Various results about rings.- 1. More examples of unique factorization domains.- 2. Ring theoretic constructions.- 3. Number theoretic applications.- Exercises.- §3. Modules.- 1. Basic facts about modules.- 2. Free modules.- 3. Integral elements of a ring.- 4. Unimodular sequences of polynomials.- §4. Algebras over a field.- 1. Definitions and examples of algebras.- 2. Division rings (skew fields).- 3. Group algebras and modules over them.- 4. Non-associative algebras.- Exercises.- Appendix. The Jordan normal form of a matrix.- Hints to the exercises.