Information about the First Qualifying Exam

 

Here is basic information about the First Qualifying Exam:

  • What is it
  • Who should take it
  • When it is given
  • Syllabus
  • How to prepare

The First Qualifying Exam is a three-and-a-half-hour written examination based on a syllabus covering introductory real variables at the level of MTH 431-432, introductory abstract algebra at about the level of MTH 419, and linear algebra at about the level of MTH 420. The examination will be given twice a year, during the week prior to the beginning of each semester.

The purpose of the first examination is to assist the Director of Graduate Studies and the student in deciding soon after the student's entry into Graduate School, whether or not he should be in the Ph.D. program.

Normally, to remain in the Ph.D. program, a student is required to pass this examination within his first two years of graduate work. A student who entered with a deficiency is not required to pass this examination until the first opportunity after he has completed two semesters of graduate work.

Syllabus for the First Qualifying Examination (Revision 01/18/12)

I. Analysis

The real number system, Least Upper Bound Axiom, limits, lim sup and lim inf, sequences and series of real numbers.

Topology of R^n including compactness, connectedness, completeness, Cauchy-Schwartz inequality, distance and norm. Functions of one and several variables, continuity, uniform continuity. Differentiability in one and several variables, derivatives and applications, partial derivatives and Jacobian matrix, chain rule, change of variables. Implicit and inverse function theorems, multivariate differential calculus. Sequences and series of functions, pointwise convergence, uniform convergence, power series, Taylor's Theorem with remainder.

Riemann Sums, Riemann integral in one and several variables, improper integrals, change of variables and Jacobian determinant. Iterated and multiple integrals, interchange of order of integration. Arc length and surface area, line integrals, surface integrals. Gauss, Green's, Stokes', and Divergence theorems.

References:

  1. R. G. Bartle & D. R. Sherbert, Introduction to Real Analysis
  2. M. Spivak, Calculus
  3. W. Rudin, Principles of Mathematics
  4. J. Munkres, Analysis on Manifolds

II. Algebra (including linear algebra)

Groups, decomposition of a group into cosets with respect to a subgroup, group homomorphisms, cyclic groups, order and index, normal subgroups, kernel, range, isomorphism, automorphisms. Fundamental theorems for homomorphisms, Fundamental Theorem of Abelian Groups. Rings, modules, and homomorphisms between them. Ideals in a commutative ring (with 1). The residue class ring R/I with respect to an ideal I.

Minimal and maximal ideals, prime ideals. Quotient field of an integral domain. The construction of polynomial rings over a commutative ring. Unique factorization domains, principal ideal domains, Euclidean algorithm. algebraic field extensions, characteristic.

Vector spaces, matrices, linear independence, spanning sets, bases, dimension, quotient vector spaces, direct sum of vector spaces, inner product spaces, orthogonality, linear transformations (vector space homomorphisms), linear functionals, kernels (null spaces), range, rank, bilinear and multilinear forms, the determinant. eigenvalues (characteristic values), eigenvectors (characteristic vectors), dual space, minimal polynomial, diagonalization, Jordan canonical form, nilpotence.

References:

  1. Herstein, Topics in Algebra
  2. Fraleigh, A First Course in Abstract Algebra
  3. Hoffman & Kunze, Linear Algebra
  4. J. Gallian, Contemporary Abstract Algebra