1. Numbers and arithmetic.
2. For what numbers n is it hard to find a factor?
3. Euclidean Algorithm, p. 41–43.
4. Rational versus integral solutions.
5. Chap. 6: Polynomial Equations.

2. Syllabus Segment 2 : Weeks 4 to 8: Chaps. 7–10: Geometry/algebra of equations MATH 4820/5820 – 3

1. Chap. 7: Analytic Geometry
• p. 112: Rotations put quadrics in standard form:
• p. 113: special curves of degree 3:
• § 7.5 p. 118: Observations on solving equations in one variable:
2. Chap. 8: Projective Geometry
• p. 130: Creating points at 1:
• p. 131: Alberti's viewpoint, p. 134: Invariance of cross-ratios:
• § 8.5: The real projective plane R², § 8.7: Homogenous coordinates:
• Bezout's Theorem, § 10.5 on Fractional power series

3. Syllabus Segment 3 : Weeks 8 to 12: Parts of Chaps. 10, 11, and 14; parts of 15, 19 and 22; then parts of 21 and 24: Complex solutions of equations and finding rational solutions among them: MATH 4820/5820 – 3

1. Rational functions and joining the end of Syllabus Part 2
2. Equations and their groups, and the use of irreducible polynomials for defining the group, and also for defining genus
3. The effect of groups in creating areas of mathematics
Syll3Fall2018-HistofMath.pdf


These will eventually contain the separate Problems set assignments for Fall 2018 Quarter:



4. 1st Problem Set: 2 lines in R³ and finding particular continued fraction: We expand last Friday's lecture on lines in R² to consider lines and planes in R³. This plays into discussing how perspective and projectivity geometry works. Then, we solve the problem in the text of finding the continued fraction expansion of the squareroot of 3, by first finding the quadratic irrationality with continued fraction expansion [t₁,t₂,t₁,t₂,…]. 1stProbset.pdf

5. 2nd Problem Set:

1. Motivation for the F(undamental) T(heorem) of A(lgebra), which reminds of topics in class.
2. Problem 1: Cotes' formula: I outline doing a problem that is in the text that appeared in integrating the function 1/(1-xⁿ).
3. Problem 2: Fractional power series for P(y) - x=0, with P a monic polynomial. This started with Newton, but became a serious tool in many studies of curves in the projective plane, especially for Riemann.
2ndProbset.pdf

6. Final Exam: (with answers) for History of Math 4820. FinalExam12-16-18.pdf



These will eventually contain the procedures and assignments for class projects:



7. 1st Potential Project: relate Euclidean Algorithm and Pell's Equation: The Euclidean algorithm applied to (r₀,r₁) is equivalent to looking at the continued fraction expansion of r₀/r₁. Between Baskara and Lagrange, that leads to always being able to solve x²-Ny²=1. A project would consider what this is a special case of, and why it was important. 1stProj-Pell.pdf

8. 2nd Potential Project: Projective Geometry: This project will use problems from the book to enhance the following topics.

1. The road to Projective Geometry: An expanded review of relating Euclidean to Projective geometry.
2. A reason for Projective Geometry: How adding points at ∞ and introducing homogenous coordinates can make seemingly different figures in the plane very much alike.
2ndProj-ProjGeom.pdf

9. 3rd Potential Project: Polynomial version of Euclid's algorithm, and how it relates to Bezout's Thm.: This project will use problems from the book to enhance the following topics.

1. Simplying the Euclidean Algorithm for polynomials.
2. How much we can learn from the case P is a quadric polynomial.
3. Reminders on linear algebra and Cramer's Rule.
4. Using the quadric case to do the analog of the general resultant.
3rdProj-Bezout.pdf

10. 4th Potential Project: Galois's Group attached to an irreducible polynomial P(y) over Q

1. The smallest field containing all the roots of P(y).
2. The definition of Field automorphisms and G_P.
3. When G_P is abelian.
4. The primitive element theorem.
4thProj-SymmGalGp.pdf

11. 5th Potential Project: The genus of a curve, a relation between Theorems of Abel, Galois and Riemann.: This project uses Puiseux expansions, that they give a field similar to the complex numbers, and that from them we produce permutations of the roots of P(x,y) with y given as functions of x.

1. A homogeneous p(x,y,z) defines a map C_p→ CP¹.
2. The Puiseux series on C_p for x'
3. Between Galois and Riemann
4. Definition of the genus of p
5thProj-TheGenus.pdf

12. 6th Potential Project: There are diophantine equations over Z for which having a Z solution is undecidable.

1. Consider Galois's Problem.
2. The notion of a diophantine set.
3. The role of Pell's equation.
6thProj-Godel-Turing.pdf


Project Summary Notes: We have six projects, and for each, while listening to them, I will be adding comments that should make an appearance in the appropriate report of your team. Some of the comments consists of observations that should help you understand the point of the project. Some of those comments could be questions on the final.



13. ProjectSummaryTopics: There are six projects. These extra comments are foremost meant for the team presenting the topic. As teams finish I will add to the pdf file, until all projects are, by the end of the semester, represented.

The point of my doing this is to show that there are connections between all of the topics. It is those connections that help make the material more memorable than they would otherwise have been. Some of the quick questions could well appear on the final. Of course, you can ask me about such questions in class. ProjectSummaryTopics.pdf

14. projprojreportslist-fall18: This list includes the project reports from each student in the class. projreportslist-fall18.html



Course Administrative Matters:



15. Required Syllabus Statements:

1. Accommodation for Disabilities: If necessary, submit your accommodation letter from Disability Services to me.
2. Classroom Behavior: in particular refers to my use of your name.
3. Honor Code: All incidents of academic misconduct will be reported to the Honor Code (honor@colorado.edu); 303-492-5550).
4. Sexual Misconduct, Discrimination, Harassment and/or Related Retaliation, refers to the Office of Institutional Equity and Compliance.
5. Religious Holidays: I will adjust to the best of my ability to your religious and family related needs.
RequiredSyllStatementsFall2018.pdf