### Math Studies Algebra, Fall 2018

From symmetry groups to fields

© 2015 Laure Bukh

Used with permission

#### When:

Mondays, Wednesdays, Fridays 10:30#### Where:

Porter Hall 226A [map]#### Recitations:

Tuesdays 19:30, Wean Hall 5310#### 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 Fridays 1:30–3:30am in Wean 6202. I am also available by appointment.

Christopher Cox will lead the recitation sessions. Most recitations will be used for presenting supplementary material. He has office hours on Tuesdays 3:30–4:30 and Thursdays 1:30–2:30 in Wean Hall 6201.

#### 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 normally 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.

The homework will count for 30% of the grade. During the semester there will be two in-class tests (on **October 3**, **November 21****November 14**).
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 10:30am of the day it is due. For each *minute* that it is late, the assignment 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 27: Introduction. Groups. Associativity as a consequence of function composition. Examples of groups. Symmetric groups. Symmetries.
*GL*(_{n}**R**). Dihedral group as symmetries of an*n*-gon. - August 28 (recitation): Mattress flipping, definition of a cyclic group, nonzero elements of
**Z**/*n***Z**are a group under multiplication iff*n*is prime. - August 29: Generators for the dihedral groups. Normal form for the dihedral group. Group presentations (somewhat informal). Order of a group. Order of an element. Cyclic groups. Fields.
*SL*(_{n}**F**). General symmetric groups. Homework #1 - August 31: Intersection and unions of subgroups. Cycles. Cyclic decomposition. Homomorphisms. Isomorphisms.
- September 3:
*Labor Day* - September 4 (recitation): Game of Nim. Direct sums and products of
**Z**/*2***Z**. - September 5: Group actions. Translation. Conjugation. Isometry group of the Euclidean plane. Automorphisms. Endomorphisms.
*Guest lecture by James Cummings* - September 7: Orbits. Fixed points. Stabilizers.
*G*-invariant sets. Cosets. Index.*Guest lecture by James Cummings*Homework #2 - September 10: Normal subgroups. Factor groups. First isomorphism theorem.
*Guest lecture by Chris Cox* - September 11 (recitation): Third isomorphism theorem. Normalizers. Product of groups.
- September 12: Index-2 subgroups are normal. Cauchy's theorem for abelian groups. Orbit-stabilizer theorem. Centralizers.
*Guest lecture by Chris Cox* - September 14: Center. Class equation. Cauchy's theorem. Generating subgroups by subsets. Maximal subgroups. Homework #3
- September 17: Axiom of choice. Well-ordering principle. Zorn's lemma. Notes on the Axiom of Choice
- September 18 (recitation): Every vector space has a basis. Continuum-sized chains in
*2*^{N}. Well-ordered Zorn. Infinite hat guessing game. - September 19: Axiom of Choice implies Zorn (finish). Partition principle. Maximal subgroups in finitely generated groups.
- September 21: Sign of a permutation. Alternating group. Alternating group is generated by
*3*-cycles. Homework #4 - September 24: Alternating group is simple. Cayley's theorem. First Sylow's theorem.
- September 25 (recitation): Another proof of Cauchy's theorem. Burnside's lemma. Applications of Burnside's lemma
- September 26: Second and third Sylow's theorems. Abelian groups are product of their Sylow subgroups. Groups of order
*pq*(part I). - September 28: Semidirect products. Groups of order
*pq*(part II). Rigid motions of**R**^{n}. Homework #5 - October 1: Words. Group words. Free groups. Reduced words.
- October 2 (recitation): Sundry group theory.
- October 3:
**Test** - October 5: Presentations. Universal property of free group. Homework #6
- October 8: Presentation of
*D*in detail. A few highlights of combinatorial group theory. Rings. Examples of rings._{2n}*M*(_{n}*R*). - October 9 (recitation): Banach–Tarski paradox.
- October 10: Zero divisors. Units. Polynomial rings. Polynomials vs polynomial maps. Power series. Laurent series. Group rings. Ring homomorphisms. Kernels. Ideals.
- October 12: Quotient rings. Ideals generated by subsets. Ideals, rings, fields generated by a set. Operations on ideals. Maximal ideals. Prime ideals. Homework #7
- October 15: Rings of polynomials in several variables. Product of ideals. Categories. Examples of categories. How to stop worrying and love the universes. Products. Uniqueness of products. Opposite category. Coproducts.
- October 16 (recitation): GCD notation. Olson's theorem.
- October 17: Examples of products and coproducts. Skeleton categories. Functors. Fundamental groups (without proofs).
- October 19:
*Mid-semester break* - October 22: Rings of fractions (localization). Universal property of localization. Homework #8
- October 23 (recitation): Division rings. Wedderburn's theorem.
- October 24: Chinese remainder theorem. Chinese remainder theorem and Lagrange interpolation formula. Euclidean domains.
- October 26:
*Classes cancelled by the president* - October 29: GCD. PIDs. Ascending chain condition. Noetherian rings. Homework #9
- October 30 (recitation): Monsky's theorem.
- October 31: Unique factorization domains. PIDs are UFDs. Greatest common divisor. Polynomial rings over UFDs (part I).
- November 2: Polynomial rings over UFDs (part II). Sums of two squares (part I).
- November 5: Sums of two squares (part II). Irreducibility criteria. Homework #10
- November 7: Eisenstein's criterion. Cyclotomic polynomials of prime order. Hilbert's basis theorem.
- November 9: Monomial orderings. Division algorithm. Gröbner bases. Monomial ideals.
- November 12: Syzygies. Buchberger's criterion. Homework #11
- November 13 (recitation): Finite fields have prime power size. Sundry ring theory.
- November 14:
**Test** - November 16: Buchberger's algorithm. Gröbner bases can be huge. Minimal Gröbner bases. Reduced Gröbner bases.
- November 19: Elimination ideals. Computing elimination ideals via Gröbner bases. Intersection of ideals. Why groups were invented (aside). Computing LCM of multivariate polynomials. Homework #12
- November 20 (recitation): Hilbert's theorem on finiteness of invariants.
- November 21:
*Thanksgiving (part I)* - November 23:
*Thanksgiving (part II)* - November 26: Radical ideals. Algebraically closed fields. Fundamental theorem of algebra. Affine varieties.
- November 27 (recitation): Finite field Kakeya.
- November 28: Extension problem. Sylvester matrix. Resultants.
- November 30: Resultants of multivariate polynomials. Extension theorem.
- December 3: Weak Nullstellensatz. Strong Nullstellensatz.
- December 5: Algebro-geometric correspondence.
- December 7: Zero-dimensional varieties. Bezout's inequality.
- December 10–16:
**Take-home final exam.**Final exam rules