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Kies de Nederlandse taal
Course module: 202001362
202001362
Algebra
Course info
Course module202001362
Credits (ECTS)3.5
Course typeStudy Unit
Language of instructionEnglish
Contact persondr. J.W. Polderman
E-mailj.w.polderman@utwente.nl
Lecturer(s)
PreviousNext 4
Lecturer
dr.ir. P. van 't Hof
Lecturer
dr. J. de Jong
Lecturer
dr.ir. R. Langerak
Examiner
dr. G. Loho
Lecturer
dr. L. Pehlivan
Academic year2021
Starting block
2A
RemarksMinor students: register for the minor!
Application procedureYou apply via OSIRIS Student
Registration using OSIRISYes
Aims
After completion of the course, the student is able to
  • Apply the (extended) Euclidean algorithm to integers and polynomials.
  • recognise and apply properties of groups
  • to derive the symmetry group of a given set and to calculate the number of orbits with the aid of Burnside’s Theorem.
  • work with ring and integral domains
  • recognise under what conditions an integral domain is a field
  • to work with the concept of vector space over an arbitrary field
  • work with the classification of finite fields
  • work with simple examples of linear code
  • investigate and understand when two algebraic structures are isomorphic
  • use and explain all components of RSA public key encryption, including all algorithmic underpinnings
Content
This course provides an introduction to abstract algebra and some of its applications. Three families of algebraic structures are introduced and studied: groups, rings and fields. Roughly speaking groups are generalisations of the set of integers equipped with addition. Rings are generalisations of the set of integers equipped with both addition and multiplication. Fields are generalisations of the set of rational numbers equipped with addition, multiplication and division by nonzero numbers. The generalisations go as far as finite structures and structures in which the operation(s) are non-commutative. The backbones of the course are permutations, finite fields. Permutations and symmetries are used in certain counting problems, such as in how many essentially different ways, taking into account the symmetries, can we paint the faces of a cube using three colours. The theory of groups and rings is used to analyse a specific public key crypto system known as RSA. Also the algorithmic principles behind RSA are treated, e.g. the Euclidean algorithm and modular exponentiation. As an application of the theory of finite fields, a brief introduction to finite codes is provided.

Topics covered in this course are:
  • Integers and integers module n
  • Group Theory, Orbits, Stabilizers
  • Cosets & Lagrange Theorem
  • Counting problems: Burnside’s theorem
  • Rings and Fields
  • Polynomial Rings
  • Finite Fields
  • Linear codes
  • RSA Public Key Encryption
Module
Module 7
Participating study
Bachelor Applied Mathematics
Participating study
Bachelor Technical Computer Science
Required materials
Book
Contemporary Abstract Algebra, Joseph A. Galian, 8th ed., Cengage Learning, 2012. ISBN: 978-1133599708
Recommended materials
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Instructional modes
Assessment

Assignment

Lecture

Tutorial

Tests
Algebra

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Kies de Nederlandse taal