Algebra is the study of operations, rules and procedures to solve equations. The origin of the term ‘Algebra’ seems to go back to a IX Century treaty by an arab mathematician with the title ‘The Compendious Book on Calculation by al-jabr and al-muqabala’. The term *al-jabr* is used in this book to denote two procedures: (i) the sum of two positive quantities to both sides of an equation, in order to cancel negative terms and (ii) the multiplication of both sides of an equation by a positive number to cancel fractions. With the passage of time, the term *al-jabr* or *algebra* became synonymous of the general study of equations and operations on them.

Algebra is one of the pillars of Mathematics and this course gives a thorough introduction to the basic concepts of Algebra.

**Lecturer:**Rui Loja Fernandes**Email:** ruiloja (at) illinois.edu**Office:** 346 Illini Hall**Office Hours:** MW 11:00-11:50 AM (or by appointment);**Class meets:** MWF 9:00-9:50 AM, 445 AH**Prerequisites:** Restricted to Graduate Students; Undergraduate students may register with approval.

**In this page:**

- Announcements
- Syllabus
- Textbooks
- Grading Policy and Exams
- Homework Assignments
- Emergency information for students in Mathematics courses

**Announcements:**

- I have posted the grades of the final. To check how you did relative to your class mates, check the graph with grades of final, midterms and homework. The cutoffs for the final grades are as follows: A+ > 460; 460 > A > 425; 425 > A- > 405; 405 > B+ > 370 ; 370 > B > 340; 340 > C > 290; 290 > D. You can see me to consult your final exam or discuss any issues with grades, today, Friday, December 16, at 5.00 pm, in my office. Here is a solution to the Final Exam.
- I have posted the grades of the 2nd Midterm on-line. To check how you are doing relative to your class mates, check the graph with grades of midterms and homework. Also, here is a solution to the 2nd midterm.
- I have posted the grades of the 1st Midterm on-line. To check how you are doing relative to your class mates, check the graph with grades of midterm and homework. Also, here is a solution to the 1st midterm.
- You can view the scores of homeworks and midterms on-line (please note that you will have to log in).

**Syllabus:**

**Group Theory.**Isomorphism theorems for groups and rings. Free groups, free products, generators and relations. Group actions on sets; orbits, stabilizers. Application to conjugacy classes, centralizers, normalizers. The class equation with application to finite p-groups and the simplicity of A_5. Composition series in a group. Refinement Theorem and Jordan-HÃ¶lder Theorem. Solvable and nilpotent groups. Sylow Theorems and applications.**Commutative rings and Modules.**Review of subrings, ideals and quotient rings. Integral domains and fields. Polynomial rings over a commutative ring. Euclidean rings, PID’s, UFD’s. Brief introduction to modules (over commutative rings), submodules, quotient modules. Tensor product. Free modules, invariance of rank. Torsion modules, torsion free modules. Primary decomposition theorem for torsion modules over PID’s. Structure theorem for finitely generated modules over a PID. Application to finitely generated Abelian groups and to canonical form of matrices. Zorn’s lemma and Axiom of Choice: Application to maximal ideals, bases of vector spaces.**Field Theory.**Prime fields, characteristic of a field. Algebraic and transcendental extensions, degree of an extension. Irreducible polynomial of an algebraic element. Normal extensions and splitting fields. Galois group of an extension. Algebraic closure, existence and uniqueness via Zorn’s Lemma. Finite fields. Fundamental theorem of Galois theory. Examples of Galois extensions. Cyclotomic extensions. If time permits, application of Galois theory to solution of polynomial equations, symmetric functions and ruler and compass constructions.

**Textbooks:**

- David Dummit and Richard Foote,
, John Wiley & Sons, Inc. 2004.**Abstract Algebra, 3rd Edition** - Thomas Hungerford,
, Springer-Verlag, GTM vol. 73, 2003.*Algebra*

I will also try to furnish some lecture notes.

**Grading Policy and Exams**

There will be weekly homework, 2 midterms and a final exam. All exams/midterms will be closed book.

**Homework and in class participation (30% of the grade)**: Homework problems are to be assigned once a week. They are due the following week, at the beginning of the Friday class.**No late homework will be accepted.**Only the ten best grades will count and the remaining homework grades will be dropped. Some homework problems will be discussed in class, with the participation of the students and this will be taken into account for the grade.**Midterms (40% of the grade):**The midterms will take place on**Friday, October 7**, and**Friday, November 11**, in the regular classroom (the dates are subject to change).**Final Exam (30% of the grade):**You have to pass the final to pass the course. According to the non-combined final examination schedule it will take place**8:00-11:00AM, Friday, December 16**, in the regular classroom.

**Homework Assignments and Sections covered so far:**

**Homework #0:**You should know the concepts and be able to solve the problems in this To Know List.**Homework #1:**Read sections 4.1 and 4.2 of the lecture notes and solve exercises 4.1.5, 4.1.6, 4.1.7, 4.1.8, 4.1.9, 4.1.13, 4.1.14, 4.1.15, 4.1.19, 4.1.20, 4.1.21, 4.1.22, 4.2.1, 4.2.4, 4.2.5, 4.2.8, 4.2.9, 4.2.11.**Homework #2:**Read sections 4.4 and 4.5 of the lecture notes and solve exercises 4.4.2, 4.4.3, 4.4.6, 4.4.7, 4.4.11, 4.4.12, 4.4.15, 4.5.2, 4.5.5, 4.5.10, 4.5.14.**Homework #3:**Read section 4.6 of the lecture notes and solve exercises 4.5.7, 4.5.13, 4.6.2, 4.6.5, 4.6.6, 4.6.7, 4.6.8, 4.6.13.**Homework #4:**Read sections 5.1 and 5.2 (up to Sylow I) of the the lecture notes and solve exercises 5.1.1, 5.1.2, 5.1.5, 5.1.6, 5.1.7, 5.2.1, 5.2.2, 5.2.3, 5.2.4.**Homework #5:**Read sections 5.2 and 5.3 of the the lecture notes and solve exercises 5.2.5, 5.2.7, 5.2.8, 5.2.10, 5.3.1, 5.3.2, 5.3.3,5.3.4, 5.3.5, 5.3.6.**Homework #6:**Read section 5.4 of the lecture notes and solve exercises 5.3.9, 5.3.10, 5.3.11, 5.4.1, 5.4.2, 5.4.4, 5.4.5, 5.4.6.**Homework #7:**Read sections 3.3-3.6 of the lecture notes and solve exercises 3.3.12, 3.3.13, 3.3.16, 3.4.2, 3.4.4, 3.4.5, 3.4.6, 3.4.7, 3.5.1, 3.5.2, 3.5.4, 3.5.9.**Homework #8:**Read sections 3.6 and 3.7 of the lecture notes and solve exercises 3.6.4, 3.6.6, 3.6.7, 3.6.9, 3.6.11, 3.6.12, 3.7.3, 3.7.6, 3.7.7, 3.7.9.**Homework #9:**Read sections 3.8 and 6.1 of the lecture notes and solve exercises 3.8.2, 3.8.3, 3.8.4, 3.8.6, 3.8.7, 6.1.2, 6.1.4, 6.1.6, 6.1.7, 6.1.8, 6.1.9.**Homework #10:**Read sections 6.2 and 6.3 of the lecture notes and solve exercises 6.2.1, 6.2.2, 6.2.3, 6.2.5, 6.2.6, 6.2.8, 6.2.9, 6.3.3, 6.3.4, 6.3.6, 6.3.7.**Homework #11:**Read sections 6.4-6.6 of the lecture notes and solve exercises 6.4.3, 6.4.6, 6.5.1, 6.5.2, 6.5.3, 6.5.4, 6.5.7, 6.6.1, 6.6.3, 6.6.4, 6.6.5, 6.6.6.**Homework #12:**Read sections 7.1-7.4 and 7.6 of the lecture notes and solve exercises 7.2.4, 7.2.6, 7.3.3, 7.3.4, 7.3.6, 7.4.2, 7.4.3, 7.4.4, 7.6.1, 7.6.3, 7.6.6.

**Sections of the Lecture Notes covered:** 4.1-4.2, 4.4-4.6, 5.1-5.4, 3.3-3.8, 6.1-6.6, 7.1-7.4, 7.6-7.8

(PDF files can be viewed using Adobe Acrobat Reader which can be downloaded for free from Adobe Systems for all operating systems.)

**Emergency information for students in Mathematics courses**

For important emergency information related to fires, tornados or active threats, please look at the following leaflet and emergency instructions.

*Last updated December 16, 2016*