Graduate study in algebra at Northern Illinois University

This page has no official standing, and is maintained for general information only by
John Beachy, Department of Mathematical Sciences, Northern Illinois University

Faculty members in algebra

John Beachy, 1969- (Ph.D. 1967, Indiana)
Noncommutative Algebra: noncommutative localization, quotient categories, injective modules
Eric Behr, 1993- (Ph.D. 1985, U.C.S.D.)
Ring Theory: Lie superalgebras
William Blair, 1971- (Ph.D. 1971, Maryland)
Ring Theory: Noetherian rings, embedding in Artinian rings
Harvey Blau, 1969- (Ph.D. 1969, Yale)
Finite Group Theory: representation theory, linear groups of small degree, decomposition of products of irreducible characters and of conjugacy classes
Harald Ellers, 1992- (Ph.D. 1989, Illinois)
Representations of Finite Groups: generalized block induction
David Hyeon, 2004- (Ph.D. 2001, Illinois)
Algebraic Geometry
Henry Leonard (Emeritus), 1968- (Ph.D. 1958, Harvard)
Finite Group Theory: representation theory (lifting representations), structure of complex linear groups of small degree
Donald McAlister (Emeritus), 1970-1988, 1994- (Ph.D. 1966, Queen's Univ., Belfast)
Semigroup Theory: algebraic structure of semigroups, regular and inverse semigroups (in particular)
Joseph Stephen, 1987- (Ph.D. 1987, Nebraska)
Semigroup Theory, Automata and Languages: combinatorial problems, the Burnside problem for semigroups, rational subsets of groups, various categories of graphs
David Trushin, 2002- (Ph.D. 1975, Ohio State)
Algebra, ring theory: Hopf algebras

Faculty research interests, in greater detail

Faculty members in related areas

Richard Blecksmith, Computational Number Theory
John Ewell (Emeritus), Number Theory
Kitty Holland, Model Theory
Yoopyo Hong, Matrix Analysis
Dave Rusin, Algebraic Topology
John Selfridge (Emeritus), Computational Number Theory
Jeffrey Thunder, Diophantine Equations
John Wolfskill, Diophantine Analysis, Algebraic Number Theory

Undergraduate / Graduate Courses

Official Northern Illinois University Graduate Catalog

420. ALGEBRA I (3). Introduction to group theory. Properties of the integers, functions, and equivalence relations. A concrete approach to cyclic groups and permutation groups; isomorphisms and the theorems of Lagrance and Cayley. PRQ: MATH 240 or consent of department. Additional information

421. ALGEBRA II (3). Continuation of MATH 420. Homomorphisms and factor groups; introduction to commutative rings, with emphasis on polynomial rings; fields and algebraic extensions. Applications to classical geometric problems. PRQ: MATH 420 or consent of department. Additional information

423. LINEAR AND MULTILINEAR ALGEBRA (3). General theory of vector spaces, linear transformations, and matrices. Topics selected from determinants, tensor products, canonical forms, and bilinear and quadratic forms. PRQ: MATH 240 and MATH 420, or consent of department.

Graduate Courses

520. ALGEBRAIC STRUCTURES I (3). Group theory including the Sylow theorems, the basis theorem for finite Abelian groups. Polynomial rings, field theory, Galois theory, solvable groups, and solvability of equations by radicals. PRQ: MATH 421 or consent of department. Additional information

521. ALGEBRAIC STRUCTURES II (3). Ring theory including the Artin-Wedderburn theorem, the Jacobson radical. Commutative algebra, Noetherian rings, and Dedekind domains. PRQ: MATH 520 or consent of department.

   Recent history of the Math 520 - 521 sequence:

   Year       taught by       required text       recommended text

   2006-2007  Blair           None                Lang
   2005-2006  Ellers          Isaacs
   2004-2005  Beachy          Lecture notes (520) Hungerford
                              Beachy (521)
   2003-2004  Stephen         Hungerford
   2002-2003  Blau            Rotman
   2001-2002  Ellers          Dummit and Foote
   2000-2001  Blair           None                Hungerford
   1999-2000  Wolfskill (520) Dummit and Foote
              McAlister (521) Dummit and Foote
   1998-1999  Beachy          Dummit and Foote (520)
                              Beachy (521)
   1997-1998  Seelinger       Hungerford (520)
                              Beachy (521)
   1996-1997  McAlister       Isaacs
   1995-1996  Thunder         None                Hungerford
   1994-1995  Ellers          Isaacs
   1993-1994  Blair           Lecture notes       Lang
   1992-1993  Beachy          Lecture notes       Hungerford
   1991-1992  Blau, Rusin     Rotman, Garling
   1990-1991  Leonard         Grove

522. HOMOLOGICAL ALGEBRA (3). Categories and functors, projective and injective modules, complexes and homology, Ext, Tor, and dimensions. Applications to cohomology of groups and ring theory. PRQ: MATH 521 or consent of department.

Recently the course has been taught as a section of Math 620

   Semester   taught by      text 

   1999, Fall   Beachy         Rotman
   1996, Fall   Beachy         Rotman
   1993, Fall   Beachy         Rotman
   1991, Fall   Beachy         Rotman

523. MODERN APPLIED ALGEBRA (3). Concepts and techniques of modern algebra which are useful in applied mathematics. Topics covered include applications of group theory to coding, applications of lattice theory to switching theory, and aplications of ring theory to linear automata. PRQ: MATH 420 or consent of department.

620. TOPICS IN ALGEBRA (3). Content varies; may include courses in semigroup theory, finite group theory, ring theory, and homological algebra. May be repeated to a maximum total of 15 semester hours. PRQ: Consent of department.

   Semester     taught by   topic

   2006, Spring   Blair       Commutative Algebra

   2005, Fall     Beachy      Homological Algebra (Osborne)
   2003, Fall     Beachy      Homological Algebra (Rotman)
   2001, Spring   Beachy      Module Categories
   2001, Spring   Blau        Algebraic Combinatorics II
   2000, Fall     Blau        Algebraic Combinatorics I
   2000, Spring   Ellers      Representation Theory II
   1999, Fall     Ellers      Representation Theory I
   1998, Fall     Seelinger   Commutative Algebra
   1998, Spring   McAlister   Semigroups II
   1997, Fall     McAlister   Semigroups
   1997, Spring   Ellers      Representation Theory II
   1996, Fall     Ellers      Representation Theory
   1996, Spring   Blau        Algebraic Combinatorics II 
   1995, Fall     Blau        Algebraic Combinatorics I
   1995, Spring   Rusin       Elliptic Curves
   1994, Spring   Beachy      Module Categories
   1992, Spring   Beachy      Module Categories
   1990, Spring   Sigurdsson  Noetherian Rings
   1989, Fall     Beachy      Rings and Modules
   1989, Spring   Lorenz      K-theory
   1989, Spring   Leonard     Representation Theory II
   1988, Fall     Leonard     Representation Theory I
   1988, Fall     Lorenz      Homological Algebra
   1988, Spring   Blair       Noncommutative Rings
   1988, Spring   Blau        Group Characters II
   1987, Fall     Blau        Group Characters I 
   1987, Spring   Blair       Homological Algebra
   1987, Spring   Kambayashi  Algebraic Geometry
   1986, Spring   Blau        Lie Algebras II
   1985, Fall     Blau        Lie Algebras I
Math 620 has also been offered as a reading course, and recent topics include:
Character Theory, Commutative Algebra, Commutative Rings, Group Representations, Lie Algebras, Module Theory, and Noetherian Rings

Syllabus for the Ph.D. Qualifying Exam in Algebra

Part I

Groups: Groups, subgroups, normal subgroups, homomorphism theorems, Sylow theorems, structure theorem for finite abelian groups, Jordan-Holder theorem, solvable groups.

Rings: Rings, ideals, homomorphisms, field of fractions of an integral domain.

Fields and Galois theory: characteristic, prime fields, algebraic and transcendental extensions, separability, perfect fields, normality, splitting fields, Galois group, fundamental theorem of Galois theory, solvability by radicals, structure of finite fields.

Linear algebra: Linear independence, basis, dimension, direct sums, linear transformations and their matrix representations, linear functionals, dual spaces, determinants, rank, eigenvalues and eigenvectors, minimal and characteristic polynomials, canonical forms.

Part II

Rings and Modules: Modules, simplicity, semisimplicity, chain conditions, tensor products, Jacobson radical, density theorem, Wedderburn-Artin theorem, finitely generated modules over a principal ideal domain, canonical forms.

Unique factorization, Euclidean domains, principal ideal domains, polynomial rings, maximal, prime, and primary ideals, Noetherian rings, Hilbert basis theorem, Lasker-Noether decomposition, integral elements, integral closure, fractional ideals, Dedekind domains.

References

Rotman, Advanced Modern Algebra
Hungerford, Algebra
Jacobson, Basic Algebra I, II
Beachy, Introductory Lectures on Rings and Modules
Dummit and Foote, Abstract Algebra
Isaacs, Algebra: A Graduate Course

    for Part I only:
Herstein, Topics in Algebra
Beachy and Blair, Abstract Algebra
Clark, Elements of Abstract Algebra
Rotman, The Theory of Groups
Garling, A Course in Galois Theory
Curtis, Linear Algebra
Friedberg, Insel, and Spence, Linear Algebra
Hoffman and Kunze, Linear Algebra


Related pages: NIU Graduate Catalog | NIU Graduate Program | John Beachy's homepage