6 ECTS credits
150 u studietijd
Aanbieding 1 met studiegidsnummer 1010430CNR voor alle studenten in het 2e semester met een gespecialiseerd bachelor niveau.
We study the algebraic structure of some important classes of groups.
Next we give an introduction to field theory and Galois theory.
In the last part we give some applications in the areas of geometric constructions and solving algebraic equations.
The main aim is to obtain a good understanding in the structure of some classes of groups and fields. It is expected that one obtains a solid knowledge of all notions and proves. ALso one has to gain the capacity of understanding and proving related structures and results. The tutorials are an important support tool to gain such expertise.
Chapter 1: Group Theory
1.1 Actions of groups
1.2 Sylow Theorems
1.3 p-Groups
1.4 Permutation groups
1.5 Direct products of groups
1.6 Finite Nilpotent Groups 1
1.7 Groep extensions
1.8 Solvable Groups
1.9 Nilpotent Groups 2
1.10 Theorem of Jordan-Holder
1.11 Free Groups
Chapter 2: Field Theory
2.1 Definitions
2.2 Simple Extensions
2.3 Isomorphisms
2.4 Algebraic Extensions
Chapter 3: Galois Theory
3.1 Galois Extensions
3.2 Normal Extensions
3.3 Separable Extensions
3.4 A characterization of Galois Extensions
3.5 Order of the Galois Group
3.6 Fundamental Theorem
Chapter 4: Cyclotomic Extensions and geometric constructions
4.1 Cyclotomic Extensions
4.2 Geometric Constructions
4.3 Roots, radicals and real numbers
Course notes available on http://homepages.vub.ac.be/~efjesper <br /> <br />Complementary study material: <br />M. Artin, Algebra, Prentice Hall, London, <br />1991. (ISBN: 0-13-004763-5) <br /> <br />I. Martin Isaacs, Algebra, A graduate course, Brooks/Cole Publshing Company, Pacific Grove, California, 1994. <br />ISBN: 0-534-19002-2 <br /> <br />M.I. Kargapolov and Ju. I. Merzljakov, Fundamentals of the theory of groups, Springer Verlag, New York, 1979.
1. Student knows fundamental concepts and theorems of groups-, field-and Galois theory.
2. Student can apply theory on examples.
3. Student independently can understand and prove related properties and structures.
4. Student can make connections with other theories, including group theory, linear algebra, module theory.
5. Student can think in function of problem solving, both individually and in group work.
6. Student can present oral solutions of exercises.
7. Student can write a mathematical text independently on the solutions of the exercises.
8. Student can consult standard references.
9. Student can look up and solve independently exercises.
10. Student can apply the theory on geometrical constructions.
De beoordeling bestaat uit volgende opdrachtcategorieën:
Examen Andere bepaalt 100% van het eindcijfer
Binnen de categorie Examen Andere dient men volgende opdrachten af te werken:
Theoretical exam 70% (oral with written preparation)
Exercises 30%
A mark will only be given if the student particpates in all tests and exams.
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