Spring 2006

MAT2200: Groups, Fields, Rings

MAT2200: Groups, Fields, Rings

Main book:

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

Additional literature:

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.