### Algebraic Structures, Spring 2023

Group ring

© 2021 Laure Bukh

Used with permission

#### When:

- Mondays, Wednesdays, Fridays 10:00 (Section A)
- Mondays, Wednesdays, Fridays 2:00 (Section B)

#### Where:

- Porter Hall A18B (Section A)
- Posner Hall 145 (Section B)

#### What:

The aim of this course is to introduce algebraic structures that pervade mathematics: groups and rings. We will learn what they are, will see many examples, learn how to reason about them. Topics to be covered include permutation groups, abelian groups, cyclic groups, homomorphisms, quotient groups, group actions, group classification, rings, ring homomorphisms, ideals, integral domains, quotient rings, unique factorization domains, principal ideal domains, and fields.

The prerequisites are being comfortable with reading and writing proofs, and a little of bit of linear algebra.

#### Resources:

- Lecture notes by Samir Siksek. They are well-written, but contain slightly less than what we are going to cover.
*Abstract Algebra*by Dummit and Foote. This is an excellent book for self-study, both at the beginner and more advanced levels. A copy is on reserve in the library.

#### Office hours:

The office hours will be at 2:30pm–3:20pm on Thursdays and at 11am–noon on Fridays in Wean 6202. I am also available by appointment.

#### Course activities:

There will be weekly homeworks, two mid-terms and a final. The mid-terms will take place on **February 22nd** and **April 12th**.
The final exam will be on **May 1st at 8:30am** in **DH 1212**. In case of a final exam conflict, the students are required to inform me
and the other involved instructor within *one week* after the registrar publishes the final exam schedule.

Students are expected to fully participate in the class. Discussions during the lectures are encouraged.

Homework will count for 15% of the grade. The mid-terms will count for 22% each, whereas the final will count for 41%.

#### Homeworks:

Practice is an integral to learning mathematics. You are encouraged to do as much homework as possible on your own; this way you will learn more. Though collaboration is allowed, you must write the solutions yourself. Turning in solutions that you do not understand will be treated as cheating. In particular, you are allowed to use (with a citation) any source, but only if you have read and understood it.

Homework must be neat. Each word must be readable. Anything that you do not want to be graded must be *completely* crossed out.
If in doubt, either re-write solution from scratch or typeset it in LaTeX. Any solution that fails to be neat will receive 0.

The lowest homework score will not count towards the final grade.

The homework must be submitted via Gradescope.

#### Exams:

All exams are closed book.

The in-class exams for sections A and B will be different. However, to avoid misplaced expectations and additional stress, I *strongly recommend*
students in section A not to discuss the exam with students in section B.

If for unforeseeable reason you are unable to take one of the exams, contact me as soon as you are able.

#### Academic integrity:

Violations of academic integrity include, but are not limited to,

- Not writing solutions independently.
- Turning in solutions that you do not understand.
- Receiving or providing assistance during an exam.

Any violation will result in automatic grade of R for the class, and will be reported to the university.

#### Staying sane and healthy:

This is an advanced mathematics course. It is designed to challenge your brain with new and exciting mathematics, not to wear your body down with sleepless nights. Start the assignments early, and get good nutrition and exercise. Pace yourself, for semester is long. If you find yourself falling behind or constantly tired, talk to me.

#### Lectures:

- January 18: Introduction. Groups. Examples. Commutativity. Abelian groups.
*D*and dihedral groups. Symmetry._{4}*GL*(_{n}**R**). Homework #1 - January 20: Notation. Identity and inverses are unique. Subgroups. Subgroup criterion. Cyclic groups and subgroups. Symmetric groups.
- January 23: Cycles in a symmetric group. Product of disjoint cycles. Order of a group. Order of a group element. Homorphisms. Homework #2
- January 25: Isomorphisms. Examples. Basic properties of homo- and isomorphisms. Conjugation. Conjugation in a symmetric group. Automorphisms.
- January 27: Examples of homomorphisms. Heisenberg group. Kernels. Cosets. Properties of cosets.
- January 30: Lagrange's theorem. Groups of prime order. Index. Euclidean algorithm. Group (
**Z**/*n***Z**)^{*}. Fermat's little theorem. Totient function. Euler's theorem. Homework #3 - February 1: Normal subgroups. Examples. Groups generated by subsets. Index-
*2*subgroups are normal. Quotients. - February 3: Projection map. Examples. Center.
*SL*(_{n}**R**). Trivial kernels. Functions from a quotient (=universal property of the quotient). [Section A: First isomorphism theorem.] - February 6: [Section B: First isomorphism theorem.] Simple groups. Product of groups. Groups actions. Group actions are homomorphisms. Cayley's theorem. Homework #4
- February 8: Orbits. Examples. Stabilizers. Orbit-stabilizer theorem. Permutations are products of disjoint cycles.
- February 10: Transpositions. Sign of a permutation. 15 puzzle. Alternating group.
- February 13: Alternating group is generated by
*3*-cycles. Simplicity of*A*. Homework #5_{n} - February 15: Converse to Lagrange's theorem fails. Large subgroups of the symmetric group.
*p*-groups.*p*-groups have nontrivial centers. Pre-image of a subgroup. [Section B: Fixed points] - February 17:
*p*-groups have subgroups of all conceivable orders. Product of two subgroups. Sylow's theorems (statement). Normalizers. - February 20: Proofs of Sylow II and III. Transitive actions and Sylow III. Homework #6
- February 24: Sylow
*p*-subgroups of*S*for small_{kp}*k*. Groups of order*pq*(easy case). [Section A: Internal product of normal subgroups.] - February 27: [Section B: Internal product of normal subgroups.] Automorphism group. Semidirect product. Infinite dihedral groups. Groups of order
*pq*(all the cases). - March 1: Orthogonal group. Rigid motions of Euclidean space. Applications of Sylow's theorem. Classification of finite abelian groups, part I (product of Sylow subgroups). Homework #7
- March 3: Classification of finite abelian groups, part II (abelian
*p*-groups). Rings. Examples. Matrices over arbitrary rings. - March 13: Commutative rings. Center of a ring. Zero divisors. Units. Quadratic extensions of
**Q**. Polynomial rings. Group rings. [Section B: Polynomials and polynomial functions.] - March 15: Polynomial rings in several variables. Ring homomorphisms. Evaluation homomorphisms. [Section A: Polynomials and polynomial functions.] Ideals. Ideals in
**Z**. Quotient rings. Generation of ideals. Sum of ideals. Product of ideals. - March 17: Intersection of ideals vs product of ideals. First isomorphism theorem for rings. Maximal ideals. Fields of prime size. Complex numbers as a quotient. Integral domains.
- March 20: Euclidean domains. Norm in quadratic extensions. Gaussian integers. PID. Euclidean domains are PID.Homework #8
- March 22: Prime ideals. Irreducibles. Ascending chains. PIDs are UFDs (existence).
- March 24: PIDs are UFDs (uniqueness). Gaussian integers are a Euclidean domain. Factorization in Gaussian integers (part I).
- March 27: Factorization in Gaussian integers (part II). Sums of two squares in
**Z**. Fields of fractions. Homework #9 - March 29: Content of polynomials. Gauss's lemma. Polynomial rings over UFD are UFD.
- March 31: Chinese remainder theorem. Lagrange interpolation formula.
- April 3: Field extensions. Subfields. Prime fields. Field characteristic. Vector spaces over arbitrary fields. Degree. Homework #10
- April 5: Adding a root of a polynomial to a field. Minimal polynomial. Algebraic elements. Algebraic extensions. Finite extensions.
- April 7: Irreducibility of polynomials of small degree. Rational root theorem. Gauss's lemma and irreducibility. Degree of iterated extension. Algebraic numbers. Q-bar.
- April 10: Algebraically closed fields. Straightedge and ruler constructions (part I). Homework #11
- April 17: Straightedge and ruler constructions (part II). Irrationality of
*e*. Splitting fields (definition and existence).Homework #12 - April 19: Splitting fields (uniqueness). Separability. Formal derivatives. Roots of unity. [Section A: Primitive roots of unity.]
- April 21: [Section B: primitive roots of unity. Cyclotomic polynomials.] Finite fields (existence). [Section A: Finite fields (uniqueness)].
- April 24: [Section B: Finite fields (uniqueness).] Counting irreducible polynomials over small finite fields.
- April 26: Finite field containment. Cyclotomic polynomials are irreducible.
- April 28: Constructibility of regular polygons (one direction). Fermat primes. Finite fields and designs.