### Math Studies Algebra, Fall 2016

From symmetry groups to fields

© 2015 Laure Bukh

Used with permission

#### When:

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

Wean Hall 8427#### 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 25% of the grade. During the semester there will be three in-class tests (on **September 23**, **October 19**, **December 2**).
Each test will count for 15% 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%.

#### Lectures:

- August 29: Introduction. Groups. Associativity as a consequence of function composition. Examples of groups. Dihedral groups.
- August 31: Fields.
*GL*(_{n}**R**). Generators for the dihedral groups. Normal form for the dihedral group. Group presentations. Symmetric groups. Cyclic permutations. Homework #1 - September 2: Cyclic decomposition. Subgroups. Homomorphisms. Heisenberg group. Isomorphisms. Group actions.
- September 5:
*Labor Day* - September 7: Centralizers and center. Normalizers. Stabilizers. Conjugation. Conjugation in the symmetric group. Cyclic groups. Homework #2
- September 9: Subgroup generated by a subset: top-down and bottom-up approaches. Direct products and direct sums.
- September 12: Uniqueness of cyclic groups. Partially ordered sets. Zorn's lemma. Axiom of choice.
- September 14: Maximal subgroups in finitely generated groups. Normal subgroups. Quotient groups. Kernels. Cosets. Homework #3
- September 16: Normal subgroups. Projection homomorphisms. Index. Index-2 subgroups are normal.
- September 19: Normality is not transitive. Cardinality of the product of subgroups. First and third isomorphism theorems.
- September 21: Simple groups. Composition series. Jordan–Hölder theorem (statement). Sign of a permutation. Homework #4
- September 23:
**Test** - September 26: Alternating group. Alternating group is generated by 3-cycles. Alternating group is simple. 15 puzzle. More examples of group actions.
- September 28: Permutation representations. Linear representations. Orbits. Kernels and faithfulness. Subgroups of index
*p*. Stabilizers. Sizes of orbits in general, and conjugacy classes in particular. (Finite)*p*-groups. Groups of order*p*are abelian. Homework #5^{2} - September 30: Right group actions. Inner and outer automorphisms. Aut(ℤ/
*n*ℤ). Statement of Sylow theorems. Cauchy's theorem for abelian groups. - October 3: Sylow theorems (part I).
- October 5: Sylow theorems (part II). Applications of Sylow theorems. Motivation for semidirect products.Homework #6
- October 7: Sylow theorems (fixing a prior mistake). Semidirect products. Examples of semidirect products. Action of semidirect products. Rigid motions of
**R**^{n}. Groups of order*pq*. - October 10: Classification of trivial semidirect products. Free groups. Group presentations.
- October 12: Fundamental groups (without proofs). Commutators. Derived subgroups. Homework #7
- October 14: Solvable groups. Rings. Examples of rings.
*M*(_{n}*R*). - October 17: Units. Zero divisors. Polynomial rings. Power series. Laurent series. Group rings. Ring homomorphisms. Ideals.
- October 19:
**Test**Homework #8 - October 24: Quotients of rings. Ideals generated by subsets. Principal ideals.
*Guest lecture by James Cummings* - October 26: Integral domains. PID. Fields. Prime ideals. Maximal ideals. Homework #9
*Guest lecture by James Cummings* - October 28: Euclidean domains. Euclidean algorithm. Gaussian integers. Complex numbers as a quotient ring.
*Guest lecture by James Cummings* - October 31: Categories. Examples of categories. Monoids. Isomorphisms. Products.
- November 2: Coproducts. Quotient rings. Localization. Homework #10
- November 4: Universal property of localization. Ideal arithmetic. Chinese remainder theorem.
*Guest lecture by James Cummings* - November 7: Chinese remainder theorem and Lagrange interpolation formula. Unique factorization domains. PID are UFD (existence part).
- November 9: PID are UFD (uniqueness part). Sums of two squares. Homework #11
- November 11: Polynomials rings over UFD are UFD.
- November 14: Roots of polynomials over a field. Rational root theorem. Eisenstein's criterion. Cyclotomic polynomials of prime order.
- November 16: Degrees, monomials and terms. Newton polygons and irreducibility. Polynomial ideals as consequences of systems of polynomial equations. Homework #12
- November 18: Noetherian rings. Hilbert basis theorem. Monomial orderings.
- November 21: The division algorithm. Gröbner bases. Monomial ideals.
- November 23:
*Thanksgiving (part I).* - November 25:
*Thanksgiving (part II).* - November 28: Syzygies. Buchberger's criterion (part I).
- November 30: Buchberger's criterion (part II). Buchberger's algorithm.
- December 2:
**Test** - December 5: Gröbner bases can be huge. Minimal Gröbner bases. Reduced Gröbner bases. Uniqueness of reduced Gröbner bases.
- December 7: Elimination ideals. Computing elimination ideals via Gröbner bases.
- December 9: More on elimination ideals. Computing intersection of ideals. Implicitization.
- December 12–18:
**Take-home final exam.**Final exam rules