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## Rings and Fields

2019-2020

### Course Objective

* The student knows basic concepts of ring theory (ring, homomorphism,
ideal, integral domain, field, units, zero divisors, PID, Euclidean
domain, UFD) and can solve problems about and with those in explicit
situations.
* The student knows quotient rings, prime/maximal ideals, and theorems
relating to those (1st isomorphism theorem, Chinese remainder theorem,
recognizing those types of ideals from quotient rings) and can apply
those in explicit situations.
* The student can determine a factorization in certain UFDs using some
irreducibility tests.
* The student knows some elementary field theory (algebraic extensions,
degrees of extensions, splitting fields, finite fields) and can apply it
in explicit situations.

### Course Content

This course studies an important algebraic structure (called a ring),
which has an addition and a multiplication satisfying certain
properties. Rings arise in many situations, and examples include the
integers, the integers modulo n, matrix rings, and polynomial rings, but
also the set of complex numbers with integral real and imaginary parts.
As is common in algebra, by formalising the common properties we can
perform general constructions and prove general results that apply in
many contexts, and we illustrate these by working out what they say in
various concrete cases. We also study particular types of rings with
properties similar to those of the integers (division with remainder,
unique factorisation). We conclude by constructing finite fields and
some of their properties. These finite fields are used frequently in
combinatorics, and they are essential to the theory of error correcting
codes (in, for example, QR-codes or electronic train tickets).

We treat the following topics.
* Definition of rings, units, zero divisors, subring, integral domain,
field; examples.
* Ideals, generators.
* Quotient rings, isomorphism theorems.
* The Chinese remainder theorem for rings.
* Polynomial rings, roots, division with remainder (in one variable).
* Prime ideals, maximal ideals.
* Principal ideal domains, Euclidean domains, unique factorisation
domains.
* Factorisation of polynomials, Eisenstein's criterion.
* Field extensions, degree of a field extension.
* Finite fields.

### Teaching Methods

Lectures (14x2 hours) and tutorials (14x2 hours)

### Method of Assessment

Two partial exams (or a resit), and marked assignments. The assignments
in total count for 5% towards the grade, the average of scores for the
two partial exams (or the score of the resit) counts for 95%.
Alternatively, the result of the resit counts 100% if this results in a
higher score. All this is subject to the requirement that the average of
the scores for the two partial exams (or the score of the resit) is at
least 55%, otherwise the grade is capped at 5.

### Literature

David S. Dummit, Richard M. Foote, "Abstract algebra", 3rd edition
(2003), John Wiley and Sons.

### Target Audience

Bachelor Mathematics Year 2

### Recommended background knowledge

The VU courses Group Theory, and Linear Algebra. Although the precise
technical knowledge from those two courses that is needed is limited,
the (algebraic and abstract) way of thinking that is developed in them
is very important for this course.

### General Information

Course Code X_400630 6 EC P1+2 300 English Faculty of Science prof. dr. R.M.H. de Jeu prof. dr. R.M.H. de Jeu prof. dr. R.M.H. de Jeu

### Practical Information

You need to register for this course yourself

Last-minute registration is available for this course.

Teaching Methods Seminar, Lecture
Target audiences

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