## Graduate Study in Algebra

### Introduction

Algebra is one of the oldest branches of mathematics, and the study of algebra in the Department of Mathematics has traditionally been rich and strong. Such eminent mathematicians as C. A. Miller (Group Theory), R. D. Carmichael (Group Theory and Number Theory), A. B. Coble (Algebraic Geometry), and R. Baer (Group Theory Abelian Groups) began a tradition in algebra that is continued today by a strong and active group of approximately 20 mathematicians, supplemented by visitors who divide their time between teaching and research.

The research strengths of the faculty are in the theory of rings (commutative and noncommutative), the theory of groups, algebraic number theory, the representation theory of groups and algebras, and algebraic geometry. A broad range of graduate courses described below is offered to prepare graduate students for further study in algebra.

The algebraic life of the Department is very active. In addition to the courses offered there is a comprehensive seminar program, usually 4-6 seminars each week, covering almost every aspect of algebra. These are almost equally divided between current topic seminars in which current research is presented in one or two lectures and in-depth seminars in which a subject is studied through a longer series of lectures. These seminar lectures are presented by faculty members, graduate students, and visitors. Graduate students are encouraged to participate in these seminars because participation in a combination of seminars and advanced courses leads a student more quickly to the frontiers of research. Often the weekly departmental colloquium lecture is presented by a noted algebraist. The Department has a number of visiting scholars and there are frequent algebraists among them.

### Guidelines to the Algebra Program

The introduction above indicates that there is a broad program in algebra in the department. This is a brief description of the various programs offered.

The basic courses, Introduction to Abstract Algebra (Math 417) and Advanced Linear Algebra (Math 418), are advanced undergraduate courses, but graduate credit can be obtained for them. They introduce the subject and are a prerequisite for further work in algebra. However, graduate students with sufficient exposure to algebra should probably begin with Math 500 and 501.

The core graduate courses in algebra are Abstract Algebra I and II (Math 500 and 501). These courses introduce algebra in a systematic way and are recommended for every aspiring (algebra) student. They also form the basis for the Comprehensive Examination in Algebra.

After these basic courses a student may choose among the various algebra courses that are more detailed, but to some extent are introductory in character.

These are Commutative Algebra (Math 502), Group Theory (Math 503), Algebraic Number Theory (Math 530), Homological Algebra (Math 505), Group Representation Theory (Math 506), and Lie Algebras (Math 507). For example, a student interested in Representation Theory of finite groups may take all of these courses (at one time in his or her career as a graduate student) because each course contains material that has application to representation theory. These courses are described below.

Each semester the department offers algebra courses listed under the umbrella title of topics (Math 595). The contents of these courses depend upon the instructor and student demand. These courses are intended to introduce the student to current topics of research in the various areas of Algebra. Examples of such courses offered recently are: Decision Problems in Group Theory, Diophantine Equations, Chevalley Groups, Algebraic Geometry, Class Field Theory, Local Cohomology, and Smooth and Etale Extensions.

Other offerings in the Department which may interest algebra students include Category Theory, Algebraic Geometry, Combinatorics, Coding Theory, and, of course, Algebraic Topology.

Each Ph.D. candidate must pass a written Comprehensive Examination in Algebra and Analysis. The examination in Algebra is based on the material in Math 500 and 501.

After finishing course work a Ph.D. candidate is expected to pass a Preliminary Examination. This exam can be based on any (two) advanced courses (Math 502 and above), including the Advanced Topics Courses as well as on special material determined by the thesis advisor.

The Department of Mathematics has two umbrella reading courses, Math 597 and Math 599. These are available to students who want to study a special topic under the guidance of a member of the faculty. (Math 599 is reserved for those students who are writing a thesis.)

### Course Descriptions

**Math 417. Introduction to Abstract Algebra**

This first undergraduate algebra course has a dual purpose: to train the
student to write proofs and to introduce axiomatic systems (with emphasis
on groups and on rings). The course begins with elementary set theory to
guarantee that students share a common knowledge of Boolean operations,
mathematical induction, and functions. The "symmetric" groups of all permutations
of a set are investigated, and the Cayley theorem (showing an arbitrary
abstract group may be regarded as a subgroup of some symmetric group) is
proved. Groups with at most 8 elements are classified, and the isomorphism
theorems are proved. An example of a quotient group is provided by the integers
modulo n, and this leads to some elementary number theory (e.g., the "little"
Fermat theorem). Finally, one considers a second algebraic system (rings)
with emphasis on fields and rings of polynomials. One shows such rings are
principal ideal domains and proves there is unique factorization of polynomials
into products of irreducible polynomials.

**Math 418. Intro to Abstract Algebra II**

Topics include: Vector spaces, subspaces, linear independence, basis and
dimension, change of basis; Fields F, ideals in F[x], irreducibility and
unique factorization in F[x], degree of extension, construction of splitting
fields and finite fields by quotient rings of F[x]; Linear transformations,
eigenvalues and eigenvectors, Jordan canonical form, operators on inner
product spaces, dual spaces, bilinear and quadratic forms. Linear codes
or Groebner bases.

**Math 500. Abstract Algebra I**

This is the first graduate level course in algebra. It introduces and
studies the basic properties of groups, rings and fields. The goal of
the course is the fundamental theorem of Galois theory and the solutions
to the three pearls of antiquity: the quadrature of the circle, the trisection
of an angle, and the duplication of the cube. The following topics are
studied: the isomorphism theorems for groups, solvability of p-groups,
simplicity of the alternating group on at least 5 letters, Sylow theorems,
Jordan-Holder Theorem, principal ideal domains, Gauss' lemma, Eisenstein's
criterion, the fundamental theorem of Galois theory, finite fields, cyclotomic
fields, solvability of equations by radicals. Prerequisites are Mathematics
417 and 418.

**Math 501. Abstract Algebra II**

This second graduate level course in algebra is devoted mainly to ring
theory. The topics studied include: Module theory, the Hilbert basis theorem,
the Krull-Schmidt theorem, the Wedderburn theorems on semi-simple rings,
the classification of finitely generated modules over a principal ideal
domain with applications to abelian groups and canonical forms for matrices,
bilinear and quadratic forms, categories, functors and the tensor product.
This course together with Math 500 forms a foundation for the other graduate
courses in algebra. Its prerequisite is Math 500.

**Math 502. Commutative Algebra**

This course studies commutative rings and modules, prime ideals, localization,
noetherian rings, primary decomposition, integral extensions and Noether
normalization, the Nullstellensatz, dimension, flatness, Hensel"s lemma,
graded rings, Hilbert polynomial, valuations, regular rings, singularities,
unique factorization, homological dimension, depth, completion. Possible
further topics: smooth and etale extensions, ramification, Cohen-Macauley
modules, complete intersections.

**Math 503. Group Theory**

This is a survey of the principal topics in theory of groups. The course
includes topics selected from the following list: the theory of nilpotent
and solvable groups, discussions of linear groups, free groups and free
products of groups, varieties of groups, group extensions, and transfer.
A recommended prerequisite for the course is Math 500-501. Basic theorems
on groups which are essential for this course are Sylow's Theorems, Jordan-Holder
Theorem, and the Fundamental Theorem on finitely generated abelian groups.
Other than those, a basic knowledge of algebra and the mathematical maturity
expected after completing Math 500-501 are all that is required.

**Math 504. Non-commutative Rings**

This course might have as its goal the definition and proof of basic properties
of the Brauer group of a commutative ring. Such notions as the Jacobson
radical of a ring, the classification of modules over a semi-simple ring,
the structure of semisimple rings, the structure of simple algebras central
over a field, the ideal theory of maximal orders, projective and injective
modules, may be considered. This course forms a background although not
a prerequisite, for the study of representations of groups. To a large
extent commutative algebra forms the foundation for the study of solutions
of polynomial equations. Perhaps its most important role is as a tool
for areas of study related to algebraic geometry. The course in commutative
algebra covers the behavior of prime ideals under flat and integral ring
extensions, especially as related to local and graded rings. The structure
of normal and regular rings as well as their completions is explored.
The notions of dimension and depth are given extensive treatment. The
techniques of homological algebra are included.

**Math 505. Homological Algebra**

This course provides the algebraic foundation for the study of derived
functors. These arise in a number of different applications such as homology
and cohomology in algebraic topology and differential geometry cohomology
of groups, and Ext and Tor in ring theory. One begins by studying resolutions
by free modules or injective modules. Then one shows how to compute derived
functors from these resolutions. Finally one studies the abstract properties
enjoyed by derived functors in general and by various specific ones in
particular.

**Math 506. Group Representation Theory**

The course studies representation of finite dimensional vector spaces
over fields of characteristic zero. The theory of characters of such representations
is developed, including induced modules and induced characters, Clifford
theory and the Artin and Brauer induction theorems. These results are
used to prove basic results on the structure of finite groups, including
Burnside's p^{a} q^{b}-theorem, Frobenius groups, and
the Burnside Transfer Theorem.

**Math 507. Lie Algebras**

In this course the following topics are discussed: solvable and nilpotent
Lie algebras, theorems of Lie and Engel, weights, trace criteria, Weyl
complete reducibility theorem for semi-simple Lie algebras, applications,
Cartan subalgebras and their conjugacy, structure of roots, the classical
Lie algebras and their root systems, complete classification of root systems
via Dynkin diagrams, Cartan matrices and the isomorphism theorem, application
to construction of Chevalley and Weyl bases, Weyl theorem on compact real
forms, the universal enveloping algebra.

**Math 530. Algebraic Number Theory**

An algebraic number field is a finite extension of the field of rational
numbers. In algebraic number fields the familiar unique factorization
into prime numbers may not hold, so the notion of a prime ideal is introduced.
In Dedekind rings every ideal can be written umquely as a product of prime
ideals. It is shown that the set of algebraic integers in an algebraic
number field is a Dedekind ring. One studies the decomposition of prime
ideals in finite extension fields. Valuation theory is introduced along
with the relation between prime ideals and discrete valuations. Ramification
groups, the different, and the discriminant of an extension are studied.
The familiar Chinese remainder theorem reappears in the form of various
approximation theorems. Often the adele ring and idele group are studied
at this point. Dirichlet's unit theorem and the finiteness of the class
number are proved for number fields. Cyclotomic fields and Kummer extensions
provide important examples of the general theory. A course in class field
theory usually follows Math 530. The Galois extensions of a given field
with Abelian Galois group are studied. Often Gauss sums, Dirichlet L-series,
and Artin L-series with their functional equation are discussed in this
second course.