### Math Studies Algebra, Fall 2017

From symmetry groups to fields

© 2015 Laure Bukh

Used with permission

#### When:

Mondays, Wednesdays, Fridays 13:30#### Where:

Wean Hall 8427 [map]#### What:

Algebra is the art of changing the perspective. The change is mainly achieved through abstraction, which strips the irrelevant details and brings the important to the forefront. The extra generality also enables the connections between far-flung mathematical concepts.

The aim of this course is both to introduce the algebraic way of thinking, and to convey the basic language of algebra. That language is the language of groups, rings, modules, fields. We shall see, for example, how the group theory unifies such topics as integer arithmetic, tessellations, solubility of polynomial equations, and counting holes in a pretzel. We shall also learn and use some category theory, which is a higher-level abstraction that unifies different algebraic notions.

#### Resources:

The book for the course is *Abstract Algebra*, 3rd ed., by Dummit and Foote. A copy of the first edition is on reserve in the library.

Not all topics that we cover are in the book, and some topics we will cover differently.

Links to additional resources will be posted as the course progresses.

#### More fun:

More fun can be had at my *office hours* on Wednesdays 2:30–3:30pm and Thursdays 9:30–10:30am in Wean 6202. I am also available by appointment.

#### Course activities:

Mastery of any subject requires practice. Hence, there will be regular homeworks. For your own good, you are strongly encouraged to do as much homework as possible individually. Collaboration and use of external sources are permitted, but must be fully acknowledged and cited. All the writing must be done individually. Failure to do so will be treated as cheating. Collaboration may involve only discussion; all the writing must be done individually. The homeworks will be returned one week after they are due.

Students are expected to fully participate in the class. The main advantage of a class over just reading a textbook is the ability to ask questions, propose ideas, and interact in other ways. In particular, discussions during the lectures are encouraged.

In the fall semester, the homework will count for 30% of the grade. During the semester there will be two in-class tests (on **September 29**, **November 6**).
Each test will count for 20% of the grade. A take-home final exam will count for 30%. No collaboration or use of external resources is allowed on tests or on the final.

Homework must be submitted in LaTeX via e-mail. I want both the LaTeX file and the PDF that is produced from it. The filenames must be of the form `lastname_alg_homeworknumber.tex`
and `lastname_alg_homeworknumber.pdf` respectively. Pictures do not have to be typeset; a legible photograph of a hand-drawn picture is acceptable.

The homework must be submitted by 1:30pm of the day it is due. For each *minute* that it is late, the grade will be reduced by 10%.

#### Staying sane and healthy:

This is an honors 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:

- August 28: Introduction. Groups. Associativity as a consequence of function composition. Examples of groups. Symmetric groups. Symmetries.
*GL*(_{n}**R**). - August 30: Dihedral groups. Generators for the dihedral groups. Normal form for the dihedral group. Group presentations (somewhat informal). Cyclic groups. Order of a group. Order of an element. Homework #1
- September 1: Homomorphisms. Isomorphisms. Heisenberg group. Subgroups.
- September 4:
*Labor Day* - September 6: Centralizers and center. Normalizers. Stabilizers. Conjugation. Cyclic permutations. Cyclic decomposition. Homework #2
- September 8: Direct sums and products. Generating subgroups by subsets.
- September 11: Uniqueness of cyclic groups. Posets.
- September 13: Zorn's lemma. Axiom of choice. Maximal subgroups in finitely generated groups. Homework #3
- September 16: Proof of Zorn's lemma. Notes on the Axiom of Choice.
- September 18: Normal subgroups. Quotient groups. Kernels. Cosets. Index.
- September 20: Index. Examples of quotients. First and third isomorphism theorems.
- September 22: Normality is not transitive. Index-2 subgroups are normal. Cardinality of the product of subgroups. Simple groups. Jordan–Hölder theorem (statement). Homework #4
- September 25: Sign of a permutation. Alternating group. Alternating group is generated by 3-cycles. 15 puzzle.
- September 27: Alternating group is simple. Permutation representations. Linear representations. Orbits. Homework #5
- September 29:
**Test** - October 2: Subgroups of index p. Stabilizers. Sizes of orbits in general, and conjugacy classes in particular. (Finite)
*p*-groups. Groups of order*p*are abelian. Right group actions.^{2} - October 4: Inner and outer automorphisms. Aut(ℤ/
*n*ℤ). Statement of Sylow theorems. Cauchy's theorem for abelian groups. Homework #6 - October 6: First Sylow's theorem.
- October 9: Second and third Sylow's theorems. Groups of order
*pq*. Recognizing direct products. - October 11: Semidirect products. Rigid motions of
**R**^{n}. Homework #7 - October 13: Words. Group words. Free groups. Universal property of a free group.
- October 16: Reduced words. Presentations.
- October 18: Rings. Examples of rings.
*M*(_{n}*R*). Units. Zero divisors. Polynomial rings. Homework #8 - October 20:
*Midsemester break* - October 23: Polynomial rings. Power series. Laurent series. Group rings. Ring homomorphisms. Ideals.
- October 25: Quotient rings. Evaluation homomorphism. Polynomials vs polynomial maps. Operations on ideals. Homework #9
- October 27: Ideals generated by subsets. Ideals, rings, fields generated by a set. Complex numbers as a quotient ring. Prime ideals. Integral domains.
- October 30: Categories. Examples of categories. How to stop worrying and love the universes. Products. Uniqueness of products. Opposite category. Coproducts. Coproducts.
- November 1: Products and coproducts in the category of abelian groups. Products and coproducts in the category of groups (without proofs). Free products. Fundamental groups (without proofs). Rings of fractions (localization). Homework #10
- November 3: Universal property of localization. Chinese remainder theorem. Chinese remainder theorem and Lagrange interpolation formula.
- November 6:
**Test** - November 8: Euclidean domains. Gaussian integers. Principal ideal domains. Prime elements and irreducible elements (part I). Homework #11
- November 10:
*No class* - November 13: Prime elements and irreducible elements (part II). Divisibility. Associates.
*Guest lecture by James Cummings* - November 15: Unique factorization domains. Noetherian rings. Noetherian induction.
*Guest lecture by James Cummings* - November 17: PIDs are UFDs. Greatest common divisor. Polynomial rings over UFDs (part I).
*Guest lecture by James Cummings* - November 20: Polynomial rings over UFDs (part II). Irreducibility criteria (part I). Homework #12
- November 27: Irreducibility criteria (part II). Cyclotomic polynomials of prime order. Polynomial ideals as consequences of systems of polynomial equations. Hilbert's basis theorem.
- November 29: Monomial orderings. The division algorithm. Gröbner bases. Monomial ideals.
- December 1: Syzigies.
- December 4: Buchberger's criterion. Buchberger's algorithm.
- December 6: Gröbner bases can be huge. Minimal Gröbner bases. Reduced Gröbner bases. Uniqueness of reduced Gröbner bases.
- December 8: Elimination ideals. Computing elimination ideals via Gröbner bases.
- December 11–17:
**Take-home final exam.**Final exam rules