### Math Studies Algebra, Spring 2018

From symmetry groups to fields

© 2015 Laure Bukh

Used with permission

#### When:

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

Scaife Hall 208#### 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.

The division of mathematics into subfields such as "algebra" is arguably quite artificial. As a believer in the unity of mathematics, I will be making frequent detours into analysis, number theory and geometry.

#### Resources:

The book for the course is *Abstract Algebra*, 3rd edition, by Dummit and Foote.

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:

The regular *office hours* for the course are Monday/Wednesday/Friday 11:30am–12:30pm in Wean 6130. I can usually meet by appointment at other times with 24 hours notice,
send me an email to make an 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 spring semester, the homework will count for 30% of the grade. There will be a take-home midterm counting for 30% and a take-home final counting for 40%.

Homework must be typeset (preferably using LaTeX) and submitted in PDF via the Canvas site. The filenames must be of the
form `andrewID_alg_homeworknumber.pdf`. 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 grade will be reduced by 10%.

#### Course information:

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.

Students with disabilities: If you have a disability and require accommodations for this course, you should contact the Director of Disability Resources and get an accommodations letter. If you have an accommodations letter from the Disability Resources office, I encourage you to discuss your accommodations and needs with me as early as possible. I will work with you to ensure that accommodations are provided as appropriate.

Stress: CMU is sometimes a stressful place. If you or anyone you know is experiencing stress, anxiety or depression I strongly encourage you to seek help and support. The university's Counseling and Psychological Services (CAPS) offers confidential counseling services and an emergency 24/7 hotline.

#### Lectures:

For reference, here is the schedule of lectures from the first semester.- Wed Jan 17: Administrivia. Recalling some ring theory. Definition of a left R-module. Linear combinations. Some examples: R^n, any left ideal of R, abelian groups (aka Z-modules)
- Fri Jan 19: Linear maps. Kernel and image of a linear map. Submodules. Span of a subset. Hom(M, N) is always an abelian group, and has an R-module structure for commutative R. Direct sum of two modules.
- Mon Jan 22: Quotient of a module by a submodule. The quotient HM. The first IM theorem. If M is a submodule of N, every HM which vanishes on M factors through the quotient HM. Linear independence for a subset of a module.
- Wed Jan 24: Basis for a module. Modules with basis are free. Structure of a free module, the construction of Fr_R(X) as function from X to R which are zero outside a finite set. If Y is a basis for N, N is isomorphic to Fr_R(Y). Quick review of category theory: objects, morphisms, functors.
- Fri Jan 26: Making a poset into a category. Viewing a commutative diagram as a functor. Natural transformations between functors. the category of functors from category C to category D. The coproduct of two objects in a category. Direct sum of two modules is a coproduct.
- Mon Jan 29: Review of initial objects and isomorphism in a category. Any two initial objects are uniquely isomorphic. Verification that direct sum is a coproduct in category of left R-modules. Opposite of a category. Hom(M, -) as a functor from R-Mod to R-Mod when R is commutative.
- Wed Jan 31: A VS is a k-module where k is a field. In a VS, bases are maximal independent sets and (Zorn's Lemma) every independent set extends to a basis. Goal: Any two bases have the same size.
- Fri Feb 2: Start of the proof that any two bases have the same size. By Cantor-Schroeder-Bernstein, enough to build injection from basis B to basis C. Given bases B and C, build an injection from B to C by replacing elements of C by elements of B, maintaining hypothesis that modified set is a basis. It is easy when B is finite, and easy-ish when B is countable as long as you make sure to consider all elements of B.
- Mon Feb 5: Finishing proof from Friday (rather informal). When B is uncountable, need to run a "transfinite" proof. Now we need to make sure that at at each "limit stage" elements of C listed so far are in the span of elements of B listed so far.
- Wed Feb 7: Goal: Theory for modules over a PID (includes VS/field and abelian groups as special cases). Any left R-module is a quotient of a free left R-module. A module is fg if it has a finite generating set, and in this case is isomorphic to a quotient of R^n. A module over an ID n has rank n if n is the max size of an independent set. Using field of fractions and a bit of VS theory, if R is an ID then R^n is a module of rank n.
- Fri Feb 9: We will show that if R PID, N free of rank n, M submodule of N then M is free of of rank m <= n and there are "nicely aligned bases" x_1, ... x_n for N and y_1,... y_m for M in the sense that y_i = lambda_i x_i for nonzero lambda_i in R such that lambda_i divides lambda_{i+1}. Since every fg module is a quotient of a free module this gives us a structure theorem for fg modules over PID.
- Mon Feb 12: If N is free of rank n then Hom(N, R) is free of rank n. The basic construction: N free of rank n, M nonzero submodule, use that R is Noetherian to find phi in Hom(N, R) which maximises the ideal phi[M]. Choose c generating phi[M] and y in M such that phi(y) = c. Argue c | psi(m) for all psi in Hom(N, R), find x in N such that c x = y. Argue that M is internal direct sum of ker(phi) intersect M and submodule generated by y, similarly N is internal direct sum of ker(phi) and submodule generated by x.
- Wed Feb 14: Finishing proof: M is free by indn on rank of M, and the aligned bases exist by induction on the rank of N. An R module M is cyclic if M = R m for some m in M. Then M is isomorphic to R/I where I is the annihilator of m, that is {r : r m = 0}. In particular M is isomorphic to R/I for some I, and convsersely R/I is cyclic because R/I = R(1 + I). The theorem from Monday shows that every fg module over PID is isomorphic to a direct sum of cyclic R-modules, of course in a PID cyclic R-modules have form R/(a). We will apply this to VS V over field k and linear map T from V to V, which can be viewed as k[x]-module via setting x V = T V.
- Fri Feb 16: V FDVS over field k and T from V to V linear. Make V into a k[x] module. By structure thm it is isomorphic as a k[x]-module to a direct sum of cyclic modules. k[x] has infinite dimension as VS over k so actually V is isomoprhic to a direct sum of modules of form k[x]/(f_i) where f_i in k[x] is nonzero and f_i divides f_{i+1}. We can drop terms where f_i is a unit.
- Mon Feb 19: Matrix of a linear transformation wrt given bases. If f is monic of degree n then k[x]/(f) has dim n as a k-VS and set of x^j + (f) for j < n forms a k-basis. The matrix of "multiply by x" wrt this basis is the companion matrix of f.
- Wed Feb 21: The Chinese Remainder Theorem as an isomorphism theorem for rings and for modules. Algebraically closed fields.
- Fri Feb 23: If k is an ACF then k[x]/(f) decomposes as a direct sum of modules of form k[x]/(x-lambda)^n. In the natural basis using powers of (x-lambda), multiply by x now has a matrix which is a Jordan block. Jordan normal form. Bilinearity. Bilin(M times N, P) is isomorphic to Hom(M, Hom(N, P)). In VS case bilinear map on U times V is determined by its restriction to product of bases
- Mon Feb 25: Definition and construction of the tensor product.

#### Homework sets etc:

- Homework 1 in PDF and TeX. Homework 1 solutions.
- Homework 2 in PDF and TeX. Homework 2 solutions.
- Homework 3 in PDF and TeX. Homework 3 solutions.
- Homework 4 in PDF and TeX. Homework 4 solutions.
- Homework 5 in PDF and TeX. Homework 5 solutions.
- Midterm solutions.
- Homework 6 in PDF and TeX. Homework 6 solutions.
- Homework 7 in PDF and TeX. Homework 7 solutions.
- Homework 8 in PDF and TeX. Homework 8 solutions.
- Homework 9 in PDF and TeX.