Lecture Course Algebra MN3

Lecturer

Anton Setzer,
House 2, Room 138,
Tel. 018 4713284,
$Replace '_dot_' by '.' and '_at_' by '@' in 'a_dot_g_dot_setzer_at_swan_dot_ac_dot_uk'$

Plan

We will follow very closely the course book [Fr99].
The following sections will be covered in the course (sections in in parenthesis will be covered if time permits):
1.1 - 1.5, 2.1 - 2.4, 3.1 - 3.3, 5.1 - 5.6, 6.1 - 6.2, 7.1 - 7.2, 8.1, 8.3, 8.5, 9.1 - 9.4, 9.6

More precise plan for the first lectures:

2. 9. Binary Operations (1.1), isomorphic binary structures (1.2)
3. 9. Groups (1.3), subgroups (1.4)
6. 9. Cyclic groups and generators (1.5)
7. 9. Groups of permutations (2.1.), Orbits, cycles and the alternating groups (2.2)
9. 9. Cosets and the theorem of Lagrange (2.3)
10. 9. Direct products and finitely generated abelian groups (2.4)
13. 9. Homomorphisms (3.1), factor groups (3.2)
16. 9. Factor group computations and simple groups (3.3)
17. 9. Series of groups (3.4), rings and fields (5.1)
20. 9. Integral domains (5.2), Fermat's theorem, Euler's theorem (5.3)
21. 9. Quotient fields (5.4), polynomial rings (5.5)
23. 9. Factorizations of polynomials over a field (5.6)
24. 9. Homomorphisms and factor rings (6.1), Prime and maximal ideals (6.2)
1. 10 Unique factorization domains (7.1), Eucledian domains (7.2)
4. 10. Introduction to extension fields (8.1), Algebraic Extensions (8.3)
5. 10. Finite fields (8.5), automorphisms of fields (9.1)
7. 10. The isomorphism extension theorem (9.2)
11. 10. Splitting fields (9.3), Separable extensions (9.4)
12. 10. Galois theory (9.6)
15. 10. Illustrations of Galois theory (9.7)

Recommended Examples

We recommend to do as many of the simpler examples from the book [Fr99] (those not marked by "theory") as possible, and as well some marked by "theory".
Some slightly more difficult but very interesting examples (examples to be discussed in the example classes are in boldface:

Section 1.1: 37
Section 1.2: 25, 26, 27
Section 1.3: 27, 28, 30, 33, 37, 38
Section 1.4: 49, 53, 54, 57
Section 1.5: 55, 56, 62
Section 2.1: 38 - 41 (with proofs; for which A,B do you get a group, for which not?); 45, 46
Section 2.2: 24 a, b (but n at least 3), 26, 31
Section 2.3: 28, 35, 45, 46 (45, 46 are a little bit longer)
Section 2.4: 48, 52
Section 3.1: 46, 49, 50, 53
Section 3.2: 24, 31
Section 3.3: 37 (relatively long), 38, 39, 40
Section 3.4: 20, 22
Section 5.1: 32, 37, 42, 46
Section 5.2: 21, 22, 26, 28
Section 5.3: 25, 26
Section 5.4: 12, 13
Section 5.5: 24, 30, 31 c
Section 5.6: 32, 34
Section 6.1: 17, 20, 28, 34
Section 6.2: 24, 28, 30, 31, 32, 34
Section 7.1: 25, 26, 30, 32, 33, 34
Section 7.2: 7, 18
Section 8.1: 28, 32, 33
Section 8.3: 21, 24, 26, 29, 34
Section 8.5: 12, 15
Section 9.1: 32, 34, 35
Section 9.2: 9, 12
Section 9.3: 17, 18, 19, 20, 21, 22, 23, 24
Section 9.4: 9, 12, 14
Section 9.6: 17, 18, 19, 21, 22, 26

Coursebook

[Fr99] John B. Fraleigh: A first course in abstract algebra, 6th edition, Addison-Wesley, 1999.

2. 9.	Binary Operations (1.1), isomorphic binary structures (1.2)
3. 9.	Groups (1.3), subgroups (1.4)
6. 9.	Cyclic groups and generators (1.5)
7. 9.	Groups of permutations (2.1.), Orbits, cycles and the alternating groups (2.2)
9. 9.	Cosets and the theorem of Lagrange (2.3)
10. 9.	Direct products and finitely generated abelian groups (2.4)
13. 9.	Homomorphisms (3.1), factor groups (3.2)
16. 9.	Factor group computations and simple groups (3.3)
17. 9.	Series of groups (3.4), rings and fields (5.1)
20. 9.	Integral domains (5.2), Fermat's theorem, Euler's theorem (5.3)
21. 9.	Quotient fields (5.4), polynomial rings (5.5)
23. 9.	Factorizations of polynomials over a field (5.6)
24. 9.	Homomorphisms and factor rings (6.1), Prime and maximal ideals (6.2)
1. 10	Unique factorization domains (7.1), Eucledian domains (7.2)
4. 10.	Introduction to extension fields (8.1), Algebraic Extensions (8.3)
5. 10.	Finite fields (8.5), automorphisms of fields (9.1)
7. 10.	The isomorphism extension theorem (9.2)
11. 10.	Splitting fields (9.3), Separable extensions (9.4)
12. 10.	Galois theory (9.6)
15. 10.	Illustrations of Galois theory (9.7)