| Announcement: The final exam will be held in East Hall Room 4 on Friday, December 19, from 13:00 to 15:00
Solutions to the second midterm and practice problems for the final exam are posted. Final exam will cover everything we have studied (see week by week detais for the exact list of topics). |
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Classes: Mo 11:15-12:30 am, Th 8:15-9:30 am
East Hall 4 Instructor: Valentina Kirichenko Office: Research I Room 125 phone: 0421-200-3184 e-mail: v.kiritchenko at jacobs-university dot de Office Hours: Mo 9:15-10:30am, Th 10:45-12:00am and by appointment |
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Teaching Assistant: Todor Milev Office: Research I Room 129A phone: 0421-200-3582 e-mail: t.milev at jacobs-university dot de Office Hours: Tu,Th 5-6pm |
There are three copies available in the library on reserve.
You might also find helpful the following notes for this
course by Michael Stoll.
Another good source is "Algebra" by Michael Artin. There are some copies in the library and I have my own copy
which you can borrow.
Group theory: Definitions and key examples. Cosets and Lagrange's theorem. Group morphisms and basic constructions including quotient groups, direct and semi-direct products and exact sequences. Group actions and orbit-stabilizer theorem. Sylow theorems.
Commutative Rings: Definitions and elementary properties. Ideals, ring morphisms and quotient rings. Domains, Euclidean domains, principal ideal domains and unique factorization.
Modules: Definitions and basic constructions. Free modules and torsion-free modules. Structure theory for finitely generated modules over a principal ideal domain.
| September 1, 4 Homework 1 due September 11 |
Set theory, equivalence relations; Complex numbers: polar form, solving equations x^n=a. Read Sections 0 and 1 in the textbook. |
| September 8, 11 Homework 2 due September 18 |
Equivalence classes; Arithmetics modulo n; Unit circle and roots of unity;
Binary structures, identity element, inverse element,
isomorphism of binary structures; Groups and monoids, subgroups, group homomorphisms. Review Sections 0 and 1. Read Sections 2, 3, 4 and 5. |
| September 15, 18 Homework 3 due September 25 |
Cyclic groups; Direct product of groups; Structure theorem for finitely generated Abelian groups
(without proof yet); Free groups, representation of groups by generators and relations, Cayley graphs. Read Sections 6, 7, 11, 39, 40. |
| September 22, 25 Homework 4 due October 2 |
Left and right cosets, Lagrange theorem; Normal subgroups, quotient groups by normal subgroups; Examples
with quotient groups of the free Abelian group of rank two; Center of a group.
Read Sections 10, 13, 14, 15 |
| September 29, October 2 Homework 5 due October 9 |
Homomorphism theorem; Groups of permutations; Cayley theorem; Symmetric groups, cycle decomposition,
conjugacy classes.
Review Sections 14,15. Read Sections 8, 9, see also articles in Wikipedia on Partition function and Young diagrams. |
| October 6, 9 Practice problems Homework 6 due October 20 |
Signature of a permutation, alternating group; Groups of symmetries
of regular polygons and polyhedra; Solving equations in radicals using symmetric groups, formula for
the roots of a cubic polynomial; Reveiw Sections 8,9. See also articles in Wikipedia on Tetrahedral symmetry, Icosahedral symmetry, Octahedral symmetry, Cubic equation, Quartic equation |
| October 13, 16 Homework 7 due October 27 |
Group actions: orbits, stabilizers,
Burnside's lemma; Automorphism groups, inner automorphisms of a group. Sylow's theorems
and applications.
Read Sections 16,17. See also articles in Wikipedia on Burnside's lemma |
| October 20, 23 Homework 8 due November 3 |
Proof of Sylow's theorems; Rings: definitions and examples, rings of integer, rational, real and complex
numbers, ring of congruence classes modulo n, ring of matrices.
Read Sections 36, 18. See also an article in Wikipedia on Sylow's theorems |
| October 27, October 30 Homework 9 due November 10 |
Division ring of quaternions; Ring homomorphisms: evaluation homomorphisms of rings of polynomials;
Characteristic of a ring; Integral domains, field of fractions of an integral domain; Finite fields,
multiplicative groups of finite fields; Ideals and quotient rings; Ring homomorphism theorem;
Coprime ideals and Chinese Remainder Theorem. Read Sections 19, 21, 22, 24. |
| November 3, 6 |
Principal Ideal Domains, Euclidean domains; Examples: Gaussian integeres,
ring of formal power series; Prime and maximal ideals; Prime and irreducible elements;
Decomposition into primes and irreducibles; Noetherian rings. Read Sections 25, 26, 27, 45, 46. |
| November 10, 13 Homework 10 due November 17 Practice problems Solutions to the first midterm |
Unique Factorization Domains; Quotients by prime and maximal ideals;
Ideals in the ring of Gaussian integers and representation of primes by sums of two squares;
The ring of polynomials over UFD is a UFD; Euclidean algorithm for Euclidean domains;
Classification of finite fields. Read Sections 45, 47. |
| November 17, 20 Homework 11 due November 24 |
Splitting fields of polynomials: existence and uniqueness;
Proof of the classification theorem for finite fields; Modules: definitions and examples. Read Sections 13.6, 12.1 in Artin. |
| November 24, 27 Homework 12 due December 4 |
Free modules; Invertible operators on free modules; Hamilton-Cayley Theorem;
Structure theorem for finitely generated modules over PIDs; Corollaries: classification of
Abelian groups, Jordan Normal Form for linear operators. Read Sections 12.2, 12.6, 12.7 in Artin |
| December 1, 4 Practice problems Solutions to the second midterm |
Noetherian modules; Submodules of free modules over PIDs, reciprocal bases;
Proof of structure theorem for finitely generated modules over PIDs; Field extensions, degree of a
field extension; Impossibilty results for constructions using compass and straightedge (without proof).
Read Sections 12.4, 12.5, 13.1, 13.2, 13.3, 13.4 in Artin |