MMA 501
Höstterminen 2013
Mälardalens Högskola
Akademin för utbildning, kultur och
kommunikation
Erik Darpö, Lars Hellström, Sergei
Silvestrov
STUDY GUIDE FOR THE COURSE ABSTRACT ALGEBRA
1. Objectives
Algebra is one of the fundamental branches of modern mathematics. It has its origins in the
theory of numbers and geometry. The aim of this course is to explore, through examples and
theory, a few of the fundamental algebraic structures underpinning modern number theory and
geometry: groups, rings and fields. We shall also see how these structures are applied in multiple
contexts such as counting and enumeration problems, coding theory and combinatorial designs.
2. Course content
Sets, equivalence relations. Groups: subgroups, permutation groups, cyclic groups, cosets, direct
product, Abelian groups, homomorphisms, quotient groups, simple groups. Rings and fields:
integral domains, ideals, homomorphims and quotient rings. Maximal ideals, polynomial rings,
factorization, field of fractions of an integral domain. Field extensions: algebraic extensions,
constructibility. Finite fields. Coding theory.
3. Course litterature
The lectures will roughly follow the presentation in
J. B. Fraleigh, A first course in abstract algebra. Addison-Wesley Publishing Co., 2003,
xiv+520 pp.
Most introductory books on abstract algebra cover the essentials of the course content, though
the order and focus of the exposition may be different. Therefore, Fraleigh is recommended as
the primary option. Some alternative choices are listed below.
P. B. Bhattacharya et al., Basic abstract algebra. Cambridge University Press, 1986,
xviii+454 pp.
J. R. Durbin, Modern algebra: an introduction. John Wiley & sons, Inc., 1992, xvi+348 pp.
T. W. Hungerford, Algebra.
xxiii+502 pp.
Reprint of the 1974 original.
Springer-Verlag, 1980,
P. A. Svensson, Abstrakt algebra. Studentlitteratur, 2001, vi+632 pp.
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4. Forms of instruction
The course will consist of fourteen lectures and four seminars. The seminars will be used for
presentation and discussion of student projects and, if need be, complementary lectures or
problem solving sessions.
5. Outline
The following is a preliminary plan for the lectures and seminars, which may be changed during
the course.
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Introduction; sets and relations;
some examples of groups;
permutations of finite sets;
groups: definition, examples;
subgroups;
cosets, direct products;
morphisms, factor groups;
rings: definition, morphisms;
seminar 1;
domains, the theorems of Fermat and Euler;
quotient fields;
ideals, factor rings;
field extensions;
automorphisms of fields extensions;
finite fields;
seminar 2;
seminar 3;
seminar 4.
6. Examination
The examination will be in the form of written homework assignments, and a student project.
Three assignments will be handed our during the course, each covering roughly one third of the
course content. The assignments are due on the 26th Nov, 6th Dec and 9th Jan, respectively.
For students projects, each student will be asked to prepare a part of the course content to
present it at the blackboard during one of the four seminars. At each presentation, one or several
students will be responsible for chairing the discussion and asking questions. This activity is
part of the examination.
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7. Teachers, contact details
Lectures will be given by Erik Darpö and Lars Hellström. Examiner is Sergei Silvestrov.
Telephone numbers:
Erik Darpö
Lars Hellström
Sergei Silvestrov
E-mail:
021-101454
021-107354
021-101524
firstname.surname@mdh.se
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