Qualifiers: Algebra
Explore Previous Algebra Qualifying Exams
Algebra Definitions
Groups:
Homomorphism theorems, permutation groups, automorphisms, finitely generated Abelian groups, products of groups, group actions, Sylow theorems, p-groups, nilpotent groups, solvable groups, normal and subnormal series, Jordan-Hölder Theorem, special subgroups (e.g., commutator subgroup, Frattini subgroup, etc.).
Rings:
Matrix rings, polynomial rings, factor rings, endomorphism rings, rings of fractions, localization and local rings, prime ideals, maximal ideals, primary ideals, integral domains, Euclidean domains, principal ideal rings, unique factorization domains, Jacobson radical, chain conditions, modules, factor modules, irreducible modules, Artinian and Noetherian rings and modules, semisimplicity.
Fields:
Algebraic extensions, algebraic closures, normal extensions and splitting fields, separable and purely inseparable extensions, theorem of the primitive element, Galois theory, finite fields, cyclotomic extensions, cyclic extensions, radical extensions and solvability by radicals, transcendental extensions.
Linear Algebra:
Matrix theory, eigenvalues and eigenvectors, characteristic and minimal polynomials, diagonalization, canonical forms, linear transformations, vector spaces, bilinear forms, inner products, inner product spaces, duality, tensors.
Suggested Courses:
- MATH 61051 / 71051: Abstract Algebra I
- MATH 61052 / 71052: Abstract Algebra II
Suggested References:
- N. Jacobson, Lectures in Abstract Algebra, Vols. I, II, III, D. Van Nostrand Co
- N. Jacobson, Basic Algebra I and II, Freeman
- S. MacLane and G. Birkhoff, Algebra, Chelsea
- T. Hungerford, Algebra, Springer-Verlag
- S. Lang, Algebra, Addison-Wesley
- M. Hall, The Theory of Groups, Macmillan
- J. Rose, A Course on Group Theory, Cambridge University Press
- J. Rotman, The Theory of Groups: An Introduction, Allyn and Bacon
- E. Artin, Galois Theory, University of Notre Dame Press
- Hoffman and Kunze, Linear Algebra, Prentice-Hall