- 1. Polynomial Rings, Ideals, and Varieties
- 1.1. Fields
- 1.2. Ideals in Polynomial Rings
- 1.3. Affine varieties
- 1.4. The ideal of a variety
- 1.5. The Algebra-Geometry Dictionary
- 1.6. Irreducibility and decomposition
- 1.7. Chain conditions
- 1.8. Operations on Ideals
- 1.9. Primes and Irreducibles
- 1.10. Dimension
- 1.11. Degree
- 1.12. The Twisted Cubics
- 2. Monomial Ideals, Groebner Bases, and the Division Algorithm
- 2.1. Monomial orders and leading terms
- 2.2. Monomial Ideals and Groebner Bases
- 2.3. The Ascending Chain Condition
- 2.4. The Division Algorithm
- 2.5. Algorithms
- 2.6. Code examples
- 2.7. Code for Groebner Basis
- 2.8. Exercises
- 2.9. Varieties over finite fields
- 3. Hilbert functions, dimension
- 4. Elimination and Intersection Theory
- 4.1. Intersection of two conics
- 4.2. Intersection of a conic and a quintic
- 4.3. Lines on the cubic surface
- 5. Appendices
- 6. References
- 7. Scratch
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