Algebraiska strukturer (10hp), HT 2009

Lärare: Svante Janson

The exam 11/12 is corrected.    Results

The reexam 7/4 is corrected.    Results

Kurslitteratur

• Grillet, Pierre Antoine: Abstract Algebra. Second edition. Springer, New York, 2007. xii+669 pp. ISBN 978-0-387-71567-4
  The course uses (at least parts of) the following sections in Chapters I-V: (preliminarily)

  I:	1-5
  II:	1, 3-5, 7, 9
  III:	1--5, 8, 10
  IV:	1-3, 5
  V:	1-3, 5-6, 8-9

Schema        Denna vecka

Tentamen

• 11 december 2009     Results
• 7 april 2010    Results
• 23 augusti 2010

Homework

1. 7 October.
2. 16 November.
3. 11 December.

Lectures

1. 31/8: Groups.  (§I.1), §I.2.
2. 2/9: Subgroups.  §I.3
3. 4/9: Homomorphisms, isomorphisms, normal subgroups.  §I.4
4. 14/9: PROBLEM SOLVING. §I.2: 1, 6, 9, 12;   §I.3: 1, 3, 6, 7, 9, 11;   §I.4: 8, 15, 16, 17, 18
5. 16/9:   Quotient groups, isomorphism theorems.   §I.5
6. 21/9:   Direct products.   §II.1
7. 22/9:   Group actions, conjugacy class, center, p-groups.   §II.3
8. 24/9:   Permutations, the symmetric group S_n, the alternating group A_n, p-groups.   §II.4, (II.5)
9. 28/9:   The Sylow theorems.   §II.5
10. 30/9:   Solvable groups   §II.7, II.9
11. 5/10:   PROBLEM SOLVING: §I.5: 4, 5, 9, 12;   §II.1: 5, 7, 8, 9;   §II.3: 5, 6, 7, 9;   §II.4: 2, 7, 9, 10;   §II.5: 2, 4, 6, 8, 9, 11, 13;   §II.9: 1, 2, 3.
12. 7/10:   Ring, subring, ideal, quotient ring, homomorphism   §III.1-3. Exerc. III.1.3, 1.5, 1.8, 2.4, 2.7, 3.1, 3.4, 3.5, 3.6
13. 12/10:   Domain, field, maximal ideal, unit, polynomial ring, principal ideal domain (PID)   §III.4-5. Exerc. III.4.1, 5.2,
14. 15/10:   Prime ideal, irreducible element, prime, unique factorization domain (UFD).   §III.8-10 Exerc. III.8.2, 8.3, 8.4, 8.5, 10.5.
15. 19/10:   Roots, formal derivative, more UFD, Gauss lemma, Eisenstein's criterion.   §III.5, 10.
16. 13/11:   Fields, field extensions.   §IV.1-2. Exerc. IV.1.1, 2.2, 2.3, 2.5
17. 16/11:   Algebraic extensions   §IV.3 (IV.4). Exerc. IV.3.1, IV.3.4
18. 20/11:   PROBLEM SOLVING: Homework 2 and recommenced exercises from this list.
19. 23/11:   K-homomorphisms, separable extensions.   §IV.4.3
20. 25/11:   Splitting fields, finite fields (Galois fields), normal extensions.   §V.1-2. Exercises V.1.1, 1.2, 1.4, 2.1, 2.2.
21. 27/11:   Galois extensions, Galois groups, fixed field, the fundamental theorem.   §V.3. Exercises V.3.1, 3.2,
22. 30/11:   PROBLEM SOLVING: Exercises listed above from Ch. IV and V.
23. 1/12:   Roots of polynomials, equations of degree 3 and 4.   §V.5, Lecture notes.
24. 7/12:   Cyclotomic polynomials, solvability by radicals, equations of degree 5, constructions with ruler and compass.   §V.6,8,9. (WARNING: Serious errors in Theorem 9.3 and Lemma 9.4! Correction.)
25. 9/12:   PROBLEM SOLVING: Exams from 2008 and 2009.