Theory of Algebraic Invariants

An English translation of the notes from David Hilbert's course in 1897 on Invariant Theory at the University of Gottingen taken by his student Sophus Marxen.

David Hilbert (Author), Reinhard C. Laubenbacher (Translated by), Bernd Sturmfels (Introduction by)

9780521449038, Cambridge University Press

Paperback, published 26 November 1993

208 pages
22.8 x 15.3 x 1.3 cm, 0.288 kg

"...a skillful, readable, coherent presentation, with a contemporary flavor..." The American Mathematical Monthly

In the summer semester of 1897 David Hilbert (1862–1943) gave an introductory course in Invariant Theory at the University of Gottingen. This book is an English translation of the handwritten notes taken from this course by Hilbert's student Sophus Marxen. The year 1897 was the perfect time for Hilbert to present an introduction to invariant theory as his research in the subject had been completed. His famous finiteness theorem had been proved and published in two papers that changed the course of invariant theory dramatically and that laid the foundation for modern commutative algebra. Thus these lectures take into account both the old approach of his predecessors and his newer ideas. This bridge from nineteenth- to twentieth-century mathematics makes these lecture notes a special and fascinating account of invariant theory. Hilbert's course was given at a level accessible to graduate students in mathematics, requiring only a familiarity with linear algebra and the basics of ring and group theory.

Preface
Introduction
Part I. The Elements of Invariant Theory: 1. The forms
2. The linear transformation
3. The concept of an invariant
4. Properties of invariants and covariants
5. The operation symbols D and D
6. The smallest system of conditions for the determination of the invariants and covariants
7. The number of invariants of degree g
8. The invariants and covariants of degree two and three
9. Simultaneous invariants and covariants
10. Covariants of covariants
11. The invariants and covariants as functions of the roots
12. The invariants and covariants as functions of the one-sided derivatives
13. The symbolic representation of invariants and covariants
Part II. The Theory of Invariant Fields: 14. Proof of the finitenesss of the full invariant system via representation by root differences
15. A generalizable proof for the finiteness of the full invariant system
16. The system of invariants I
I1, I2, …, Ik
17. The vanishing of the invariants
18. The ternary nullform
19. The finiteness of the number of irreducible syzygies and of the syzygy chain
20. The inflection point problem for plane curves of order three
21. The generalization of invariant theory
22. Observations about new types of coordinates.

Subject Areas: Algebra [PBF]