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Cursus: 202001362
202001362
Algebra
Cursus informatie
Cursus202001362
Studiepunten (ECTS)3,5
CursustypeOnderwijseenheid
VoertaalEngels
Contactpersoondr. J.W. Polderman
E-mailj.w.polderman@utwente.nl
Docenten
VorigeVolgende 4
Docent
dr.ir. P. van 't Hof
Docent
dr. J. de Jong
Docent
dr.ir. R. Langerak
Examinator
dr. G. Loho
Docent
dr. L. Pehlivan
Collegejaar2021
Aanvangsblok
2A
OpmerkingMinor students: register for the minor!
AanmeldingsprocedureZelf aanmelden via OSIRIS Student
Inschrijven via OSIRISJa
Cursusdoelen
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
Inhoud
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
Participating study
Bachelor Applied Mathematics
Participating study
Bachelor Technical Computer Science
Module
Module 7
Verplicht materiaal
Book
Contemporary Abstract Algebra, Joseph A. Galian, 8th ed., Cengage Learning, 2012. ISBN: 978-1133599708
Aanbevolen materiaal
-
Werkvormen
Assessment

Hoorcollege

Opdracht

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Toetsen
Algebra

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