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* Employs proven conception of teaching topics in commutative algebra through a focus on their applications to algebraic geometry, a significant departure from other works on plane algebraic curves in which the topological-analytic aspects are stressed *Requires only a basic knowledge of algebra, with all necessary algebraic facts collected into several appendices * Studies algebraic curves over an algebraically closed field K and those of prime characteristic, which can be applied to coding theory and cryptography * Covers filtered algebras, the associated graded rings and Rees rings to deduce basic facts about intersection theory of plane curves, applications of which are standard tools of computer algebra * Examples, exercises, figures and suggestions for further study round out this fairly self-contained textbook

Inhaltsverzeichnis§ 1. Derivations.§ 2. Differential Algebras.§ 3. Universal Extension of a Differential Algebra.§ 4. Description of the Universal Extension in Special Cases.§ 5. Differential Modules of Field Extensions.§ 6. Differential Modules of Local Rings.§ 7. Differential Modules of Affine Algebras.§ 8. Smooth Algebras.§ 9. Differential Modules of Complete Intersections.§ 10. The Kahler Differents (Jacobian Ideals) of an Algebra.§ 11. Universally Finite Differential Algebras.§ 12. Differential Algebras and Completion.§ 13. Differential Modules of Semianalytic Algebras.§ 14. Regularity Criteria for Semianalytic Algebras.§ 15. Existence of p-Bases.§ 16. Traces of Differential Forms.§ 17. Residues in Algebraic Function Fields of one Variable.Appendices.A. Commutative Algebras.B. Dimension Formulas in Algebras of Finite Type.C. Complete Intersections.D. The Fitting Ideals of a Module.E. The Dual of a Module over a Noetherian Ring.F. Traces.G. Differents.Symbol Index.

The book serves as a foundational textbook in its field, offering comprehensive insights and methodologies that have shaped academic discourse since its original publication in 1985. It features clear explanations, illustrative examples, and practical applications, making it an essential resource for students and professionals alike. The English translation ensures accessibility to a wider audience, preserving the original's depth while enhancing understanding for non-native speakers. This enduring work continues to influence contemporary thought and practice.