Algebra Qualifying Examination
- Fundamentals: sets, relations, and functions, Cartesian
products and operations, partial orderings, equivalence relations and
partitions, the Axiom of Choice and Zorn's Lemma, cardinal numbers.
- Integers: mathematical induction and the least number
principle, congruence, Division Algorithm, unique factorization,
greatest common divisor and least common multiple, Euclidean Algorithm.
- Groups (basic theory): semigroups and monoids, various
characterizations of groups, subgroups, normal subgroups, homomorphism,
isomorphism, quotient groups, direct products and sums, cosets and
counting, Lagrange's Theorem, subgroup generation, the Isomorphism
Theorems, the Correspondence Theorem.
- Examples of groups: permutation groups, groups of symmetries, matrix groups, dihedral and quaternion groups.
- Permutation groups: Cayley's Theorem, permutations as
products of disjoint cycles and consequences for the structure of
permutation groups, permutations as products of transpositions,
alternating groups, simplicity of An for n > 4.
- Abelian groups: structure of cyclic groups, free abelian
groups and the structure of finitely generated abelian groups, the
Fundamental Theorem of Abelian Groups.
- Groups: structure theory, group actions on sets,
stabilizers, the class equation, generalizations of Cayley's Theorem,
Cauchy's Theorem, Sylow Theorems, classification of finite groups of
- Rings (basic theory): subrings, ideals, homomorphism,
isomorphism, quotient rings, direct products and sums, isomorphism and
correspondence theorems, division rings and fields, examples of rings,
rings of endomorphisms of an abelian group, rings of matrices.
- Rings (advanced theory): properties of ideals, maximal and
prime ideals, the Chinese Remainder Theorem, integral domains,
relationship between maximal ideals and fields and between prime ideals
and integral domains, factorization in commutative rings, irreducible
and prime elements, Euclidean domains, principal ideal domains, unique
factorization domains, polynomial rings.
- Matrix arithmetic, reduced row ecehelon form of a matrix, general solution to a system of linear equations.
- Determinants and their properties.
- Vector spaces: subspaces, basis, coordinate vectors, change of basis.
- Linear transfomations: matrix of a transformation, kernel, range, rank, Dimension Theorem, linear functionals, dual basis.
- Inner products: Cauchy-Schwartz inequality, orthonormality,
Gram-Schmidt, projection, Hermitian adjoint of a matrix and
- Eigenvalues, eigenvectors, characteristic polynomial,
minimal polynomial, Cayley-Hamilton Theorem, algebraic and geometric
- Unitary matrices and transformations, normal matrices and
transformations, unitary diagonalization of normal matrices, Spectral
Theorem, Schur's Unitary Triangularization Theorem.
- Canonical forms: Jordan canonical form, rational canonical
form, invariant factors, elementary divisors, Primary Decomposition
- Hermitian matrices, Rayleigh-Ritz Theorem, variational
characterization of eigenvalues (min-max) and applications,