Lectures in Abstract Algebra

1. Extension of homomorphisms.- 2. Algebras.- 3. Tensor products of vector spaces.- 4. Tensor product of algebras.- I: Finite Dimensional Extension Fields.- 1. Some vector spaces associated with mappings of fields.- 2. The Jacobson-Bourbaki correspondence.- 3. Dedekind independence theorem for isomorphisms of a field.- 4. Finite groups of automorphisms.- 5. Splitting field of a polynomial.- 6. Multiple roots. Separable polynomials.- 7. The “fundamental theorem” of Galois theory.- 8. Normal extensions. Normal closures.- 9. Structure of algebraic extensions. Separability.- 10. Degrees of separability and inseparability. Structure of normal extensions.- 11. Primitive elements.- 12. Normal bases.- 13 Finite fields.- 14. Regular representation, trace and norm.- 15. Galois cohomology.- 16 Composites of fields.- II: Galois Theory of Equations.- 1. The Galois group of an equation.- 2. Pure equations.- 3. Galois’ criterion for solvability by radicals.- 4. The general equation of n-th degree.- 5. Equations with rational coefficients and symmetric group as Galois group.- III: Abelian Extensions.- 1. Cyclotomic fields over the rationals.- 2. Characters of finite commutative groups.- 3. Kummer extensions.- 4. Witt vectors.- 5. Abelian p-extensions.- IV: Structure Theory of Fields.- 1. Algebraically closed fields.- 2. Infinite Galois theory.- 3. Transcendency basis.- 4. Lüroth’s theorem.- 5. Linear disjointness and separating transcendency bases.- 6. Derivations.- 7. Derivations, separability and p-independence.- 8. Galois theory for purely inseparable extensions of exponent one.- 9. Higher derivations.- 10. Tensor products of fields.- 11. Free composites of fields.- V: Valuation Theory.- 1. Real valuations.- 2. Real valuations of the field of rational numbers.- 3. Real valuations of ?(x) which are trivial in ?.- 4. Completion of a field.- 5. Some properties of the field of p-adic numbers.- 6. Hensel’s lemma.- 7. Construction of complete fields with given residue fields.- 8. Ordered groups and valuations.- 9. Valuations, valuation rings, and places.- 10. Characterization of real non-archimedean valuations.- 11. Extension of homomorphisms and valuations.- 12. Application of the extension theorem: Hilbert Nullstellensatz.- 13. Application of the extension theorem: integral closure.- 14. Finite dimensional extensions of complete fields.- 15. Extension of real valuations to finite dimensional extension fields.- 16. Ramification index and residue degree.- VI: Artin-Schreier Theory.- 1. Ordered fields and formally real fields.- 2. Real closed fields.- 3. Sturm’s theorem.- 4. Real closure of an ordered field.- 5. Real algebraic numbers.- 6. Positive definite rational functions.- 7. Formalization of Sturm’s theorem. Resultants.- 8. Decision method for an algebraic curve.- 9. Equations with parameters.- 10. Generalized Sturm’s theorem. Applications.- 11. Artin-Schreier characterization of real closed fields.- Suggestions for further reading.