# Math 500 – Abstract Algebra I – Fall 2014

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: Wed 11:00 AM (or by appointment);
Class meets: MWF 1:00-1:50 PM, 243 Altgeld Hall;
Prerequisites: Restricted to Graduate Students; Undergraduate students may register with approval.

## Syllabus:

• Group Theory. Isomorphism theorems for groups. 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. 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, Abstract Algebra, 3rd Edition, John Wiley & Sons, Inc. 2004.
• Thomas Hungerford, Algebra, Springer-Verlag, GTM vol. 73, 2003.

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 10, and Friday, November 7, 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 7:00-10:00PM, Thursday, December 18, 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 section 4.3 of the lecture notes and solve exercises 4.3.1, 4.3.3, 4.3.4, 4.3.5, 4.3.6, 4.3.8, 4.3.9, 4.3.10, 4.3.11.
• Homework #3: 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, 4.5.7, 4.5.10, 4.5.14.
• Homework #4: Read section 4.6 of the lecture notes and solve exercises 4.6.1, 4.6.2, 4.6.5, 4.6.6, 4.6.7, 4.6.8, 4.6.10, 4.6.11, 4.6.13, 4.6.14.
• Homework #5: Read sections 5.1 and 5.2 of the the lecture notes and solve exercises 5.1.2, 5.1.5, 5.1.6, 5.1.7, 5.2.3, 5.2.4, 5.2.5, 5.2.7, 5.2.8.
• Homework #6: Read sections 5.3 and 5.4 of the the lecture notes and solve exercises 5.3.1, 5.3.2, 5.3.4, 5.3.6, 5.3.9, 5.3.10, 5.3.12, 5.4.1, 5.4.2, 5.4.4, 5.4.5, 5.4.8.
• Homework #7: Read sections 3.3 and 3.4 of the lecture notes and solve exercises 3.3.3, 3.3.5, 3.3.6, 3.3.8, 3.3.10, 3.3.12, 3.3.13, 3.3.14, 3.4.2, 3.4.4, 3.4.6, 3.4.7, 3.4.8.
• Homework #8: Read sections 3.5 and 3.6 of the lecture notes and solve exercises 3.5.1, 3.5.2, 3.5.4, 3.5.5, 3.5.7, 3.5.9, 3.5.10, 3.6.3, 3.6.5, 3.6.7, 3.6.9, 3.6.11, 3.6.12.
• Homework #9: Read sections 3.7 and 3.8 of the lecture notes and solve exercises 3.7.2, 3.7.3, 3.7.6, 3.7.7, 3.7.9, 3.7.11, 3.8.2, 3.8.3, 3.8.4, 3.8.6, 3.8.7.
• Homework #10: Read sections 6.1 and 6.2 of the lecture notes and solve exercises 6.1.2, 6.1.3, 6.1.4, 6.1.6, 6.1.7, 6.1.9, 6.2.1, 6.2.2, 6.2.3,6.2.6, 6.2.8, 6.2.9, 6.2.10.
• Homework #11: Read sections 6.3 and 6.4 of the lecture notes and solve exercises 6.3.2, 6.3.3, 6.3.5, 6.3.6, 6.3.7, 6.3.8, 6.4.1, 6.4.3, 6.4.4, 6.4.5.
• Homework #12: Read sections 6.5, 6.6 and 7.1 of the lecture notes and solve exercises 6.5.1, 6.5.2, 6.5.4, 6.5.7, 6.5.8, 6.5.9, 6.6.1, 6.6.3, 6.6.4, 6.6.5, 6.6.6, 6.6.7.
• Homework #13: Read sections 7.3, 7.4 and 7.5 of the lecture notes and solve exercises 7.3.1, 7.3.3, 7.3.4, 7.3.6, 7.4.2, 7.4.3, 7.4.4, 7.5.2, 7.5.3, 7.5.6.

Sections of the Lecture Notes covered: 3.3-3.8, 4.1-4.6, 5.1-5.4, 6.1-6.6, 7.1.,7.3-7.7. (Sections 7.2 and 7.8 where mentioned in the last Lecture of the semester, without much details).