Math 104a – Number Theory (Summer 2018)


General information

Course description: This course is an introduction to number theory. The following is a tentative list of topics that will be covered: arithmetic in the integers, linear diophantine equations, arithmetic in other rings of integers, Pythagorean triples, residue systems and congruences, the Chinese remainder theorem, the structure of residue rings, basic arithmetic functions, the quadratic reciprocity law.

Lectures: MWF from 12:00 to 13:50 in AP&M 7421.

Discussion sections: TuTh from 12:00 to 12:50 in AP&M 6402.

Instructor: François Thilmany – fthilman(at)ucsd(dot)edu – AP&M 6132.

TA: Stephan Weispfenning – sweispfe(at)ucsd(dot)edu – AP&M 6434.

Textbook: William J. LeVeque's Fundamentals of Number Theory.

Announcements

  • No class on Independence Day (Wednesday 4th of July).
  • For the lectures on Pythagorean triples, I used Keith Conrad's notes on the Gaussian integers. Section 8 contains the result we have covered in class (theorem 8.3), but feel free to browse through the other sections. Some of the basic properties of Z[i] that we have discussed are also proved in Conrad's notes.
  • The midterm will be held in class on Monday 23d of July.
  • Here is a non-exhaustive list of results that were covered in the course and that you can use, if quoted appropriately, in homework and exams.
  • The final exam will be held on Saturday 4th of August, from 11:30 to 14:30 in AP&M 7421.
  • Office hours

    My office hours are held MWF after lectures, from 14:00 to 14:50 in my office (AP&M 6132) or in classroom AP&M 7218 (one floor above), if my office gets crowded.
    On Thursdays, I will extend section with another office hour from 13:00 to 13:50 in AP&M 6402.
    If these times do not work for you, please send me an email and we will set up an appointment.
    You are welcome to stop by my office, although I may not always be available.

    Stephan's Tuesday office hour will be an extension of discussion section: it will be from 13:00 to 13:50 in AP&M 6402.
    On Friday, Stephan will have an office hour in AP&M 6434 from 17:00 to 18:00.

    Evaluation policy

    Grade: The final grade will be the maximum between the weighted averages (25% homework + 25% midterm + 50% final) and (25% homework + 75% final).

    Sections: Sections are mandatory and attendance will be monitored. Results and exercises covered in section will be part of the material for the course.
    During section, you are encouraged to actively work on the problems. Stephan will be in the room to answer any questions and review parts of the material, but he will not present full solutions.
    To allow more time to work on the problems, on Tuesdays his office hours will extend the section. On Thursday, section will be extended by my office hour, in the same room (AP&M 6402).

    Homework: The problem sets and deadlines are to be found below. Late homework will not be accepted.
    You are encouraged to collaborate on the assignments, but the work you hand in must be your own.

    Midterm: The midterm will be in class, on Monday 23rd of July. There will be no make-up midterm; if you miss the midterm, its part of the grade will be replaced by the final exam. No books, notes, or electronic devices are allowed during the exam.

    Final: The final exam will be held on Saturday 4th of August from 11:30 to 14:30 in TBA. There will be no make-up final. If you miss the final exam, you will receive an 'F', unless the cause is an unpreventable emergency and justification is provided before the exam. In that case, you will receive an 'incomplete' and will be required to take the exam in the following quarter (Fall 2018). No books, notes, or electronic devices are allowed during the exam.

    General rules: The clarity of your answers are a substantial part of the evaluation, so make sure that you always write intelligibly, and always justify your answers! An illegible, disorganized or unclear work might receive no credit.
    The academic integrity rules of the university will be strictly enforced.
    Any request for a regrade must be made within a week and before the exam / homework in question leaves the instructor's or the TA's sight.

    Lectures

  • 07/02: Introduction and course administration. The set Z of integers. Divisibility, the Euclidean algorithm, gcd's, Bézout's identity, Euclid's lemma, unique factorization.
  • 07/03 (in section): Euclid's proof of the infinitude of primes.
  • 07/04: None; Independence Day.
  • 07/06: Unique factorization (continued). Computing gcd's using the Euclidean algorithm, application to linear diophantine equations.
  • 07/09: Groups, rings. Arithmetics in Euclidean domains.
  • 07/11: The Gaussian integers are a Euclidean domain, examples of factorization and primes in Z[i]. Classifying Pythagorean triples using factorization in Z[i] (notes).
  • 07/13: Classifying Pythagorean triples (continued). Ideals and quotients of rings, quotients of Z, formalism of modular arithmetic.
  • 07/16: Other examples of residue rings. Homomorphisms and isomorphisms of rings, the isomorphism theorem. Product of rings. Applying the homomorphism ZZ/mZ to study equations.
  • 07/18: Prime and maximal ideals, pairs of comaximal ideals. The Chinese remainder theorem for rings and for Z.
  • 07/20: Application of the Chinese remainder theorem: solving systems of congruences.
  • 07/23: Midterm exam. Euler's totient function φ and Euler's formula for φ.
  • 07/25: Lagrange's theorem about the order of subgroups and elements. Euler's theorem and Fermat's little theorem about the order of units of Z/mZ. Wilson's theorem.
  • 07/27: Counting elements of given order in a cyclic group, finite subgroups of the multiplicative group of a field are cyclic. Computing the order of units in Z/pnZ.
  • 07/30: The group of units of Z/mZ is cyclic when m is a power of an odd prime, the structure of the group of units of Z/mZ for arbitrary m. Euler's criterion for squares in Z/pZ, the Legendre symbol and its properties.
  • 08/01: Gauss' lemma; for which primes p is 2 a square modulo p? The quadratic reciprocity law.
  • 08/03: Applications of quadratic reciprocity. Hensel's lemma; squares in Z/mZ.

    Homework

    The problem sets are posted below. You can bring your answers in class or drop them in the box in the basement.
    Homework is due on Fridays at 18:00 (unless indicated otherwise). Please note that no credit will be given for any homework turned in late.

  • Problem set 1 is due Friday 6 July.
  • Problem set 2 is due Friday 13 July.
  • Problem set 3 is due Friday 20 July.
  • Problem set 4 is due Friday 27 July.
  • Problem set 5 is due Friday 3 August.