Algebra 611 (Fall 2009) and 612 (Spring 2010)
The class meets on MWF 10:10 - 11:00 am in LGRT 1114

Lecture notes


Here I will post remarks on the covered material, homework assignments, and lecture notes for selected classes.
Please scroll down to get to the most recent posts. I will keep 611 materials for easier reference.

Vector Spaces. 5 lectures based on Chapter II of Knapp.
Vector spaces and linear maps. Linear independence, dimension, basis. Infinite-dimensional vector spaces. Duality for vector spaces. Linear algebra in coordinates. Quotients and direct sums. Determinants.
Homework 1 due 9/21.

Inner-Product Spaces. 3 lectures based on Chapter III of Knapp.
Inner product: real symmetric and complex Hermitian. Schwartz inequality and its friends. Bessel's inequality. Gram-Schmidt orthogonalization process. Orthogonal and unitary groups. Adjoint and self-adjoint linear maps. Spectral Theorem. Polar decomposition.
Homework 2 due 9/28.

Categories and Functors. 4 lectures based on lecture notes.
Categories. Examples: Vect_k, Sets, Groups, Ab, Rings, TopSpaces (topological spaces and continuous maps). A group as a category with one object. Poset as a category. Covariant and contravariant functors. Examples: forgetful functors, duality in Vect_k, TopSpaces->Rings (algebra of functions/pull-back of functions), complexification as a functor Vect_R->Vect_C. Equivalence of categories (was defined in class as a fully faithful and essentially surjective functor). Example: equivalence of the category of finite-dimensional vector spaces to the category with one object for each natural number and morphisms given by matrices. Representable functors. Products and coproducts. Examples in Vect_k and in Sets. Example of a category without a product. Natural transformations. Example: canonical isomorphism from V to V^** as a natural transformation.
The lecture notes include the Homework 3 due 10/12.

Groups and Group Actions. 4 lectures based on IV.1,2,3,6,10 and VI.1 of Knapp.
Three lectures were devoted to the proof of Sylow theorems. Along the way we reviewed the vocabulary of undergraduate group theory: groups, group actions, counting with groups, cosets, Lagrange theorem, homomorphisms, normal subgroups, quotient groups, first fundamental theorem, second fundamental theorem, center, conjugacy classes, stabilizer, normalizer, centralizer, etc. In the last lecture free groups and subgroups of free groups were discussed.
Homework 4 due 10/26.

Rings and Modules. 3 lectures based on IV.4,VIII.1,2,4 of Knapp.
Category of commutative rings. Ideals. First isomorphism theorem for rings (If f:R->S is a surjective homomorphism of rings then S is isomorphic to the qiotient ring of R by the ideal Ker f. There is a canonical bijection between ideals of S and ideals of R containing Ker f.) Category of R-modules. First isomorphism theorem for R-modules. Direct products and direct sums (coproducts) of rings and modules. Integral domains. Principal ideal domains. Principal ideal domains are factorial. Characteristic of a field. Finite fields.
Homework 5 due 11/2.

Structure theorem for finitely generated (f.g.) modules over PID. 4 lectures based on VIII.6 of Knapp.
Second isomorphism theorem for R-modules. Free R-modules. Linear algebra of free R-modules. Finitely presented R-modules. Any f.g. module over PID admits a finite presentation. Smith normal form of a matrix over PID. Any f.g. module of a PID is a direct sum of cyclic modules. Maximal ideals. Existence of maximal ideals. If Rⁿ is isomorphic to R^m then n=m. Chinese remainder theorem for PID. Young diagrams. Uniqueness theorem for f.g. modules over PID.
Homework 6 due 11/9.

Linear Operators. 2 lectures based on Knapp V.
Equivalence of a category of k[x]-modules and a category of vector spaces with a linear operator. Proof of the primary decomposition theorem, the cyclic (or rational) normal form, and existence/uniqueness of the Jordan canonical form. Characteristic polynomial. Minimal polynomials. Cayley-Hamilton theorem.
Homework 7 due 11/16.

Bilinear Forms. 2 lectures based on Knapp VI.1, VI.2, VI.3
Bilinear form, its matrix. Change of matrix under the basis change. Radical of a bilinear form. Orthogonal complement. Principal Axis Theorem for symmetric forms. Signature of a real symmetric form. Sylvester theorem. Canonical form of an alternating bilinear form. If V has a nondegenerate alternating form then dim V is even.
Take-home midterm due 11/23.

Tensor products. 4 lectures based on lecture notes
Tensor product of vector spaces and R-modules. Application: Hilbert's 3d problem. Right-exactness of the tensor product. Restriction and extension of scalars.
The lecture notes include the homework 8 due 12/7.

Commutative Algebra. 5 lectures based on Knapp VIII.2,3,4,5,8,9
Ascending chain condition. Noetherian rings. Hilbert Basis Theorem. Field of fractions. Prime ideals. Gauss Lemma. If R is a UFD then R[x] is a UFD. Eisenstein criterion. Integral closure.
Last homework 9 due 12/11.

Final exam for 611 (Fall 2009):   Final Exam study guide   Final Exam

Field Extensions. 5 lectures based on Knapp IX.1-5 or lecture notes
Algebraic and transcendental elements. Minimal polynomial. Simple algebraic extensions K(a). Degree of the extension. Finite extensions. Adjoining roots. Splitting fields (existence and uniqueness up to isomorphisms). Algebraic closure. Finite fields. Constructions with straightedge and compass.
Homework 1 due 2/1.

Galois Theory. 3 lectures based on Knapp IX.6-8 or lecture notes
Separable extensions. Theorem of the primitive element. Normal extensions. Galois extensions. Fundamental Theorem of Galois Theory.
Homework 2 due 2/8.

Applications of Galois Theory. 5 lectures based on Knapp IX.9,10,11 or lecture notes
Fundamental Theorem of Algebra. Galois group of a finite field. Adjoining root of unity and cyclotomic fields. Kronecker-Weber Theorem: quadratic case. Cyclic extensions. Composition series and solvable groups.
Homework 3 due 2/22.

Applications of Galois Theory - II. 5 lectures based on Knapp IX.11,13,14,15 or lecture notes
Solvable extensions. Solving polynomial equations in radicals. Norm and trace. Lagrange resolvents. Solving solvable equations. Transcendental numbers. Hermite's Theorem.
Homework 4 due 3/1.

The in-class midterm on Galois Theory is scheduled for Wednesday, March 3.

Transcendental Extensions. Algebraic Sets 6 lectures based on lecture notes
Algebraic independence. Transcendence degree. Noether's Normalization Theorem. Weak Nullstellensatz. Strong Nullstellensatz. Closed algebraic sets.
Homework 5 due 3/29.

Localization. 3 lectures based on Knapp VIII.10 and lecture notes
Localization. Local rings. Ideals and prime ideals in R and in S^-1R. Nilradical. Going-up Theorem. Spec R. Examples from Algebraic Geometry and from Number Theory.
Homework 6 due 4/5.

Algebra and Geometry. 3 lectures based on Knapp VIII.10 and lecture notes
Localization of modules. Nakayama's Lemma. Irreducible algebraic sets. Spectrum of the algebra of polynomials. Morphisms of algebraic sets. Dominant morphisms. Finite morphisms. Finite morphisms are surjective and have finite fibers. Geometric form of the Noether Normalization Theorem.
Homework 7 due 4/12.

A take-home midterm due 4/21.

Representation Theory of Finite Groups. Based on Knapp VII.4 and lecture notes
Examples of representations. Category of representations. Complete reducibility: Maschke's Theorem. Schur's Lemma. One-dimensional representations. Representations of Abelian groups (dual group). Characters. Schur's orthogonality relations for characters. The number of irreps is equal to the number of conjugacy classes. The sume of squares of dimensions of irreps is equal to the size of the group. The dimension of any irrep divides the size of the group. Homework 8 due 5/3.

A final examination.

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Syllabus

Algebra is the language of several central mathematical fields (Number Theory, Algebraic Geometry, Representation Theory, and Algebraic Topology) and has applications to practically all mathematical disciplines. It describes an incredibly elegant and sophisticated world of pure mathematical abstraction, which nevertheless is surprisingly useful. Algebra is ideally suited for playing with new rules and new objects. It has a lot of flexibility and generates new approaches to mathematical theories as quickly as our immune system generates responses to new diseases.

The first semester (611) will be mostly about commutative aspects of algebra, more specifically commutative rings and modules (generalization of vector spaces). The second semester (612) will be mostly devoted to noncommutative algebra (Galois theory and basic representation theory). The concept of a commutative ring is a synthesis of two notions: a ring of algebraic numbers (number theory) meets a ring of functions (analysis). An amazing observation is that geometry of any space is encoded in algebra of its ring of functions and any ring, including rings of interest to Number Theory, can be realized in this way. This is one of the cornerstones of modern mathematics that leads to algebraization of geometry and geometrization of number theory.

The course has two goals. The first one is to introduce you to modern algebra on the level necessary for successful graduate studies. We will cover all material required for the advanced qualifying algebra exam (and for the Linear Algebra portion of the basic qualifying exam).

The second goal is to give you sufficient background in algebra to enable you to take more advanced graduate classes in algebraic disciplines and to read papers that use algebraic language. The course will also start to prepare those of you who are interested in doing research in algebra. Our department has a vibrant algebraic community represented by three research groups, Algebraic Geometry, Number Theory, and Representation Theory, and four research seminars, Five College Number Theory Seminar, Five College Valley Geometry Seminar, Reading Seminar in Algebraic Geometry, and Representation Theory Seminar. It includes Tom Braden, Eduardo Cattani, Peter Dalakov, Farshid Hajir, Paul Gunnels, Paul Hacking, Eyal Markman, Ivan Mirkovic, Peter Norman, Alexei Oblomkov, Eric Sommers, Jenia Tevelev, Tom Weston, Siman Wong, and Giancarlo Urzua.

The textbook for this class is Knapp's Basic Algebra. Be advised that this book covers the bare minimum of the necessary material many lecture topics (especially in 612) are not sufficiently covered there. I will expect you to take lecture notes. We will cover a fair amount of commutative algebra, however, this is a course in Algebra and not in Commutative Algebra. It is aimed at all graduate students, and the syllabus is a result of many compromises. People who are thinking about doing research in algebra are advised to get a supplementary textbook, ideally Introduction to Commutative Algebra by Atiyah and MacDonald. This is an amazing book that covers commutative algebra in much greater depth and detail than our course. This is a short book but it contains a treasure trove of exercises. Another amazing book is Lang's Algebra. It covers practically all basic algebra necessary for the working mathematician. There are many other nice textbooks, for example Abstract Algebra of Dummit and Foote, Artin's Algebra, and Rotman's Advanced Modern Algebra. There are also many textbooks devoted specifically to Galois Theory,

Algebra is a language, and as with any language, the key to success is practice. I will distribute homework problem sets weekly and you should expect to spend a considerable amount of time solving these problems. At the beginning, you will probably have a feeling that each problem requires its own proof strategy and that some problems are really tricky to solve. However, at some point you will start to recognize the same old tricks and methods reused over and over again. You will become an algebraist! There are certain things you can do to systematically improve your problem-solving techniques. For example, I am a big fan of active reading of mathematical textbooks. This means that each time you see a little statement, lemma, or theorem, you should try to come with its proof, or at least with the main idea of the proof, before you read the proof in the book. In fact, sometimes it is easier to come up with your own proof than to read a polished argument, because sophisticated notation can obscure a simple underlying principle. This is especially true for arguments (abundant in algebra) that amount to checking many little straightforward steps. However, some arguments are truly tricky and if your attempt to prove something on your own ends in failure, you will better appreciate someone's ingenuity.

There will be one in-class midterm, one take-home midterm, and the final. The course grade will be determined as follows: 20% Homework, 5% Class and Review Session participation, 25% In-class midterm, 25% Take-home midterm, 25% Final Exam.

I will have office hours (date/time TBA) and a weekly review session of homework problems (date/time TBA). I will discuss homework problems after you turn them in. I will try my best to set up office hours that everybody can attend, so please keep me informed about changes in your schedule that interfere with office hours.

The course webpage is www.math.umass.edu/~tevelev/611.html. Please check it often for homework handouts and other relevant information.