Spring 2006
MAT2200: Groups, Fields, Rings

Main book:
J. Fraleigh. A first course in abstract algebra. 7th edition.

S. Lang. Algebra.

Lecture 1 (16.01)
Sets and their cardinality. Relations on sets.
Sections: 0.

Lecture 2 (17.01)
Equivalence relations. Binary operations.
Sections: 0, 2, 3.
Exercises:
Section 0: 14, 17, 29, 30;
Section 1: 32, 34;
Section 2: 23;
Section 3: 2, 8, 24, 25.

Lecture 3 (24.01)
Groups. Multiplication tables.
Sections: 4, 2.
Exercises:
Section 4: 1-6, 11-18, 28, 29, 32.

Lecture 4 (31.01)
Subgroups. Homomorphisms. Cyclic groups.
Sections: 5, 13, 6.
Exercises:
Section 5: 13, 22, 23, 45, 49, 51;
Section 6: 44, 12-16, 46, 51, 52.

Lecture 5 (7.02)
Subgroups of cyclic groups. Permutation groups.
Sections: 6, 8.
Exercises:
Section 6: 33-37, 47, 56;
Section 8: 1-5, 10, 40-43, 52.

Lecture 6 (14.02)
Permutation groups. Cosets. Group action on a set.
Sections: 9, 10, 16.
Exercises:
Section 8: 36, 47;
Section 9: 1-6, 13, 27a,b;
Section 10: 15, 16, 35, 38, 40;
and these three.

Lecture 7 (21.02)
Group action on a set. Direct products of groups. Factor groups.
Sections: 16, 17, 11 (first subsection), 14.
Exercises:
Section 10: 34, 41, 46;
Section 11: 3-7, 50, 51;
Section 14: 34-36, 39.

Lecture 8 (28.02)
Factor groups. Finitely generated abelian groups. p-groups.
Sections: 14, 11, 36.
Obligatory exercises

Lecture 9 (6.03)
p-groups. Sylow subgroups.
Sections: 36, 37.

Lecture 10 (7.03)
Sylow subgroups.
Sections: 36, 37.

Lecture 11 (13.03)
Rings and fields. Integral domains. Fields of quotients.
Sections: 18, 19, 21.

Lecture 12 (14.03)
Homomorphisms of rings. Prime and maximal ideals.
Sections: 26, 27.

The corrected exercises are at the reception on the 7th floor. Here is the list of those who have passed. The next deadline is 29.03, 14-30.

Lecture 13 (27.03)
Principal ideal and unique factorization domains.
Sections: 45.

Lecture 14 (28.03)
Polynomial rings. Irreducible polynomials.
Sections: 22, 23.
Exercises:
Section 36: 13, 15;
Section 18: 24, 25, 28, 41, 52;
Section 19: 2;
Section 20: read the section, solve 4, 11, 12, 27, 28.

Lecture 15 (4.04)
Irreducible polynomials. Field extensions.
Sections: 23, 29.

Lecture 16 (18.04)
Vector spaces. Finite extensions.
Sections: 30, 31.
Exercises:
Section 21: 1, 2;
Section 22: 17, 29, 30;
Section 23: 1, 4, 12, 16;
Section 29: 1-3;
and these two.

Lecture 17 (25.04)
Algebraic extensions. Algebraically closed fields.
Sections: 31, 48, 49.

Lecture 18 (2.05)
Isomorphism extension theorem. Splitting fields. Multiplicity of zeros of polynomials.
Sections: 49, 50, 51.
Exercises:
Section 30: 2, 20, 23, 24, 25, 27;
Section 31: 1, 3, 5, 24, 28, 30;
Section 48: 3, 5, 39;
and this one.

Lecture 19 (9.05)
Separable and normal extensions.
Sections: 51, 53.
Exercises:
Section 31: 25, 36;
Section 33: 9, 13, 14;
Section 49: 4, 5;
Section 50: 1, 3, 6;
and these three.

Lecture 20 (16.05)
Main theorem of Galois theory. Cyclic extensions.
Sections: 53, 55, 56.
Exercises:
Section 15: 39;
Section 50: 1, 6;
Section 51: 1, 4;
Section 53: 7;
and these two.

Lecture 21 (23.05) (last lecture)
Radical extensions. Insolvability of the quintic.
Sections: 56.