Course syllabus for Algebra

Course syllabus adopted 2021-02-26 by Head of Programme (or corresponding).

Overview

  • Swedish nameAlgebra
  • CodeMVE150
  • Credits7.5 Credits
  • OwnerMPENM
  • Education cycleSecond-cycle
  • Main field of studyMathematics
  • DepartmentMATHEMATICAL SCIENCES
  • GradingTH - Pass with distinction (5), Pass with credit (4), Pass (3), Fail

Course round 1

  • Teaching language English
  • Application code 20126
  • Open for exchange studentsYes

Credit distribution

0107 Examination 7.5 c
Grading: TH		 7.5 c

In programmes

Examiner

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Eligibility

General entry requirements for Master's level (second cycle)
Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements above.

Specific entry requirements

English 6 (or by other approved means with the equivalent proficiency level)
Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements above.

Course specific prerequisites

Basic course in linear algebra

Aim

In swedish only

Learning outcomes (after completion of the course the student should be able to)

  • define and explain what a binary operation is.
  • define the most important algebraic structures: groups rings and fields.
  • give exmples of groups consisting of congruences of integers, matrices, permutations, and symmetries of geometric objects.
  • define what a subgroup and a coset of a subgroup is.
  • use equivalence relations to study cosets and to prove Lagrange’s theorem.
  • form quotient objects of groups and rings by means of normal subgroups and ideals.
  • define the concepts: homomorphism, isomorphism, kernel and image of a homomorphism.
  • use the Euclidean algoritm for integers and polynomials over a field and describe the corresponding theories of unique prime factorization.
  • explain the relation between finite field extensions and zeros of polynomials over the base field.

Content

Operations, groups, subgroups, symmetries, permutations, equivalence relations and partitions, prime numbers, the fundamental theorem of arithmetic, congruences, orders of groups and elements in groups, cyclic groups, cosets and Lagrange's theorem, isomorphisms, direct product of groups, isomorphism types of finite Abelian groups, Cayley's theorem, group homomorphisms, image and kernel, normal subgroups, quotient groups, the fundamental homomorphism theorem, orbits, stabilizers, Burnside's theorem, Sylow's theorem, definition of rings and fields, integral domains, characteristic of a field, polynomial rings, the division algorithm, irreducible polynomials,Euclidean rings, unique factorization domains, ring homomorphisms, ideals, principal ideals, quotient rings, adjunction of roots of polynomials, something about the existence and construction of finite fields, zeros of polynomials, factorization in polynomial rings, various number systems.

Organisation

The course consists of about 15 lectures and 15 lessons. Classes are devoted to demonstrations of the exercises in the textbook.

Literature

Durbin: Modern Algebra, John Wiley & Sons

Examination including compulsory elements

Written exam

The course examiner may assess individual students in other ways than what is stated above if there are special reasons for doing so, for example if a student has a decision from Chalmers on educational support due to disability.