Algebra
and advanced linear algebra
(BE2)
  Bachelorkursus, forår 2010 / Bachelor course, spring 2010  

Kursuslærer: Jørgen Larsen.
Kursusmateriale: Niels Lauritzen: Concrete Abstract Algebra. Cambridge University Press, 2003.
Her er en uformel note med nogle grundlæggende begreber.
Her er en engelsk-dansk ordliste.
Kursusbeskrivelsen kan ses her.
Tidspunkt: Mandag 13-16 og torsdag 9.15-12, første gang torsdag den 4. februar 2010.
Sted: Bygning 27 lokale I.

Teacher: Jørgen Larsen.
Textbook: Niels Lauritzen: Concrete Abstract Algebra. Cambridge University Press, 2003.
Here is an informal note presenting some basic concepts.
A specification of the course is found here.
Time: Mondays from 13 to 16 and Thursdays from 9:15 to 12, beginning 4th February 2010.
Place: Building 27 room I.

Eksamen er en kombineret skriftlig og mundtlig eksamen, jf. kursusbeskrivelsen. Opgaveformuleringerne afhentes hos studienævnssekretæren onsdag den 2. juni 2010 kl. 11. Besvarelserne afleveres samme sted mandag den 7. juni 2010 kl. 11.00. Den mundtlige eksamen afholdes mandag den 21. juni 2010.

The examination is a combined written and oral eksamination, cf. the specification of the course. The written set of problems can be picked up at the secretary on Wednesday 2nd June 2010, at 11 a.m., and they should be handed in at the same place not later than Monday 7th June at 11 a.m. The oral examination will take place on Monday 21rd June.

The 2009 exam problems, Danish version / English version.

Schedule

Monday 17. May:

Discussion of exercises as needed.

Monday 10. May: Ascension Day.

Thursday 13. May:

Discussion of exercises as needed.

Thursday 6. May:

We will go through last year's exam problems (Danish version / English version): Each student in turn will have the opportunity to present a sub-problem at the blackboard.

Monday 26. April:

(We will not introduce new material, but do (and discuss) some exercises.)

Thursday 22. April:

Ideals in polynomial rings (ctd.), finite fields (pp. 164-166, 170-171).

Monday 19. April:

Polynomial rings modulo ideals (pp. 164-66).
Exercises 4.26, 4,27, 4.28, 4.29, 4.30.

Thursday 15. April:

Primitive roots; ideals in polynomial rings (pp. 157-163 except subsections 4.5.1 and 4.5.2).

Monday 12. April:

Cyclotomic polynomials, roots of polynomials (pp. 150-157).
Exercises: 4.13 (where p denotes a prime number) (Hint: Use Lemma 3.5.12.), 4.8.

Thursday 8. April:

Polynomial rings, division of polynomials (pp. 143-150).
Exercises: 4.3, 4.4, 4.5.

Thursdag 1. April and Monday 5. April: Easter holidays.

Thursday 25. and Monday 29. March:

Unique factorization (pp. 125-138).
Exercises: 3.16, 3.26, 3.27.

Monday 22. March:

Ring homomorphisms, fields of fractions (pp.119-125).
Here are some Chapter 2 and 3 exercises: Danish version / English version.

Thursday 18. March:

Ideals, quotient rings, prime ideals, maximal ideals. (pp. 115-119).
Exercises: 3.12, 3.13, 3.14, 3.15.
Here are two more Chapter 2 exercises, Danish version / English version.

Monday 15. March:

Definition of a ring, a field, a domain; Gaussian integers (pp. 111-115).
Exercises 3.1 and 3.3, and these exercises.

Thursday 11. March:

Symmetric and alternating groups (pp. 78-88).
Exercises: 2.39 (+ 2.23), 2.40.

Monday 8. March:

Euler's Theorem revisited (section 2.8.1), permutations and transpositions (pp. 78-85).
Exercises from the backlog.

Thursday 4. March:

Cyclic groups.

Monday 1. March:

Group homomorphisms, the isomorphism theorem, the order of a group element (pp. 68-73).
Exercises: 2.20, 2.22, 2.23, 2.27, 2.28, 2.31, 2.32.

Thursday 25. February:

Normal subgroups, residue classes (pp. 64-67).
Exercises: 2.10, 2.11, 2.16, 2.13, 2.14, 2.15, 2.19, 2.17.

Monday 22. February:

More examples of groups. Subgroups and cosets (pp. 57-64).
Excercises: 2.4, 2.6, 2.8, 2.9.

Thursday 18. February:

Groups: definition, simple properties and some examples (pp. 50-57 top)
Exercises: 2.1, 2.2, 2.3.

Monday 15. February:

Euler's theorem, prime numbers, unique factorization (pp. 17-22).
Exercises: 1.20 and 1.21 (two "applied" problems), 1.32 (a simple fact), 1.30 (item (ii) gives a surprising formula), 1.29 (Wilson's theorem is referred to later on in the text).
Here is a short summary of Chapter 1: Danish version / English version.

Thursday 11. February:

The (extended) Euclidian algorithm, The Chinese remainder theorem (pp. 9-17).
Exercises: 1.11 (the proof that the congruence relation is an equivalence relation), 1.12 (an example), 1.14 (referred to by Remark 1.5.9), 1.17 and 1.18 (two examples).

Monday 8. February:

Division with remainder, Congruence modulo d, Greatest common divisor, The Euclidean algorithm (pp. 5-9).
Exercises: 1.3, 1.4, 1.8, 1.7, (1.1, 1.2).

Thursday 4. February:

Presentation of the course, and a gentle introduction to some of the issues of sections 1.2 - 1.4.