### General Information

Course Code | X_400630 |
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Credits | 6 EC |

Period | P1+2 |

Course Level | 300 |

Language of Tuition | English |

Faculty | Faculty of Science |

Course Coordinator | prof. dr. R.M.H. de Jeu |

Examiner | prof. dr. R.M.H. de Jeu |

Teaching Staff |
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 |
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Target audiences

This course is also available as:

### 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 assignmentsin 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 precisetechnical 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.