MM510 - Rings and number theory. Autumn 2009, 2nd quarter.


| Announcements | Contact | Course overview | Schedule | Literature | Lectures | Weekly exercise sheets | Exam | Links |

Announcements

  • There will be an extra question hour on Monday 21 December, 12-14, in U49.

  • There will be a question hour on Friday 18 December, 14-16, in U49B.

  • The second set of mandatory exercises can be found below on weekly sheet 5. Solutions should be handed in no later than Monday 7 December.

  • There will be a lecture on Wednesday 25 November, 14-16, in U48. The exercise seesion has been moved to Thursday 26 November, 10-12, in U28.

  • The first set of mandatory exercises can be found below on weekly sheet 3. Solutions should be handed in no later than Monday 23 November.

  • The first lecture will be Monday 2 November, 12-14, in U89a.

Contact

    The lecturer is Martin Svensson, and you can find my home page here. You are always welcome to come and see min in my office. If you want to be sure to catch me, I suggest you call me or send me an email first. The teaching assistant is Lena Erbs.

Course overview

    The official course description can be found here in Danish and here in English, and the syllabus can be found here. As the course name suggests, the theme for the course is the algebraic structures that generalize calculus with ordinary integers. We will therefore begin with the study of integers, and then work our way forward to rings and fields. The main topics are as follows:

    1. Integer calculus, division of integers, the algorithm of Euclid, prime numbers, prime factorization of integes.

    2. Modular arithmetic: congruence, congruence classes of integers.

    3. Rings: definition, examples and properties, integral domains, fields, homomorphisms, ideals, quotient rings.

    4. Polynomials and rings of polynomials, polynomial division, irreducible polynomials, factorization of polynomials, congruence classes of polynomials.

    5. Field extensions, vector spaces, bases (if time permits).

    I hope to cover chapters 1-6 and, if time permits, 9-10 in Hungerford's book. Interesting applications can be found in chapters 12 and 13, which we may discuss, again if time permits. I will occasionally refer to the appendices, and in general I will assume knowledge of this material.

Schedule

    The schedule for the course can be found here.

Literature

    I will use and closely follow the book by Thomas W. Hungerford, Abstract Algebra, an Introduction, Second Edition, which is available for purchase in the university book shop.

    On the reference shelf in IMADA's library you can find some additional litterature that may be of interest to you.

Lectures

    Week Section Subject
    45 1.1, 1.2, 1.3 Integers, the division algorithm, divisibility, greatest common divisor, the Euclidean algorithm, prime numbers and prime factorization.
    46 2.1, 2.2, 2.3 Congruence modulo n, congruence classes, modular arithmetics, Zp when p is prime, equations in Zn.
    47 3.1, 3.2, 3.3 Rings, integral domains, fields, units, zero divisors, subrings, homomorphisms and isomorphisms.
    48 4.1, 4.2, 4.3 Rings of polynomials, the division and Euclidean algorithms in F[x], irreducible polynomials, unique factorization of polynomials.
    49 4.4, 5.1, 5.2, 5.3 Roots, the Remainder and Factor Theorems, congruens and congruence classes modulo p(x), extension fields.
    50 6.1, 6.2 Ideals, principal ideals, congruence modulo an ideal, quotient rings, first isomorphism theorem.
    51 6.3 Prime ideals and maximal ideals.

Weekly exercise sheets

Exam

    4 hours written exam with all the standard tools (books, calculator). Grading according to the Danish 7-grade scale. External examiner. In addition to the written exam, als mandatory exercises to be handed out during the course (pass/fail, internal evaluation by the teacher). The mandatory exercises must be passed in order to attend the written exam.

Links

Last modified: Fri Jul 23 18:06:05 CEST 2010
Martin Svensson (svensson@imada.sdu.dk)