A Companion to Lang’s Algebra

"In these notes, I attempt to bring together in an orderly arrangement materials I have given to my classes over the years when teaching Berkeley’s basic graduate algebra course, Math 250, from Lang’s Algebra – motivations, explanations, supplementary results and examples, advice on material to skip, etc.. I follow this expository material with some exercises not in the text that I particularly like, together with notes on a few of the exercises in the text." Preface Notes on the Text Chapter I. Groups. Re §I.1. Monoids. Re §I.2. Groups. Re §I.3. Normal subgroups. Re §I.4. Cyclic groups. Re §I.5. Operations of a group on a set. Re §I.6. Sylow subgroups. Re §I.7. Direct sums and free abelian groups. Re §I.8. Finitely generated abelian groups. Re §I.9. The dual group. Re §I.10. Inverse limit and completion. Re §I.11. Categories and functors. Re §I.12. Free groups. Chapter II. Rings. Re §II.1. Rings and homomorphisms. Re §II.2. Commutative rings. Re §II.3. Polynomials and group rings. Re §II.4. Localization. Re §II.5. Principal and factorial rings. Chapter III. Modules. Re §III.1. Basic definitions. Re §III.2. The group of homomorphisms. Re §III.3. Direct products and sums of modules. Re §III.4. Free modules. Re §III.5. Vector spaces. Re §III.6. The dual space and dual module. Re §III.7. Modules over principal rings. Re §III.8. Euler-Poincaré maps. Re §III.9. The Snake Lemma. Re §III.10. Direct and inverse limits. Chapter IV. Polynomials. Re §IV.1. Basic properties of polynomials in one variable. Re §IV.2. Polynomials over a factorial ring. Re §IV.3. Criteria for irreducibility. Re §IV.4. Hilbert’s Theorem. Re §IV.5. Partial fractions. Re §IV.6. Symmetric polynomials. Re §§IV.7-9. Chapter V. Algebraic Extensions. Re §V.1. Finite and algebraic extensions. Re §V.2. Algebraic closure. Re §V.3. Splitting fields and normal extensions. Re §V.4. Separable extension. Re §V.5. Finite fields. Re §V.6. Inseparable extensions. Chapter VI. Galois Theory. Re §VI.1. Galois extensions. Re §VI.2. Examples and applications. Re §VI.3. Roots of unity. Re §VI.4. Linear independence of characters. Re §VI.5. The norm and trace. Re §VI.6. Cyclic extensions. Re §VI.7. Solvable and radical extensions. Re §VI.8. Abelian Kummer theory. Re §VI.9. The equation Xn − a = 0. Re §VI.10. Galois cohomology. Re §VI.11. Non-abelian Kummer extensions. Re §VI.14. Infinite Galois extensions. Chapter VIII. Transcendental Extensions. Re §VIII.1. Transcendence bases. Chapter X. Noetherian Rings and Modules. Re §X.4. Nakayama’s Lemma. Chapter XII. Absolute Values. Re §XII.1. Definitions, Dependence, and Independence. Chapter XIII. Matrices and Linear Maps. Re §XIII.1. Matrices. Re §XIII.2. The rank of a matrix. Re §XIII.3. Matrices and linear maps. Re §XIII.4. Determinants. Re §XIII.5. Duality. Re §XIII.6. Matrices and bilinear forms. Re §XIII.7. Sesquilinear duality. Chapter XIV. Representation of One Endomorphism. Re §XIV.1. Representations. Re §XIV.2. Decomposition over one endomorphism. Re §XIV.3. The characteristic polynomial. Chapter XVI. The Tensor Product. Re §XVI.1. Tensor product. Re §XVI.2. Basic properties. Re §XVI.3. Flat modules. Re §XVI.4. Extension of the base. Re §XVI.5. Some functorial isomorphisms. Re §XVI.6. Tensor product of algebras. Re §XVI.7. The tensor algebra of a module. Re §XVI.8. Symmetric products. Appendix 2. Some Set Theory. Re §A2.1. Denumerable sets. Re §A2.2. Zorn’s Lemma. Re §A2.3. Cardinal numbers. Re §A2.4. Well-ordering. Exercises Errata