# Graduate Exams: The Qualifying Exam and The Comprehensive Exam

## Ph.D. Qualifying Exam in Mathematics

The Ph.D. Qualifying Exam in Mathematics is a two-day written exam on the core topics: Abstract Algebra and Real Analysis. Exams are typically administered at the beginning of the Fall and Spring semesters.

Students are expected to take Abstract Algebra (MATH 7261-7262) and Real Analysis (Math 7350-7351) before attempting the qualifying exams. Exceptions may be granted by the Graduate Committee in very special situations.

Students wishing to take the qualifying exams should consult with their advisors who will make the necessary arrangements with Dr. Pei-Kee Lin.

The Ph.D. comprehensive exam in Mathematics consists of two parts: the completion of the qualifying exam (both the algebra and the analysis components) and a talk on a topic with potential for advanced research. This talk must be announced to the department and evaluated by a committee of at least three faculty members. Upon completion of these two parts, the comprehensive exam results form should be sent to the Graduate School.

## Qualifying Exams Spring 2025 Dates

**Real Analysis**: 9:00-12:00 Monday, Jan 13, 2024. (Room DH 249)

**Algebra**: 9:00-12:00 Wednesday, Jan 15, 2024 (Room DH 249)

Anyone who wants to take any test should register to Dr. Pei Kee Lin before Nov 19, 2024

## Qualifying Exams Fall 2025 Dates

**Real Analysis: **9:00-12:00 Monday, Aug 18, 2025. (Room: DH 249)

**Algebra: **9:00-12:00 Wednesday, Aug 20, 2025. (Room DH 249)

Anyone who wants to take any test should register to Dr. Pei Kee Lin before April 25, 2025.

### Abstract Algebra, Math 7261-7262

Background topics: Groups, subgroups, homomorphisms, Lagrange's Theorem, normal subgroups, quotient groups, isomorphism theorems, group actions, orbits, stabilizers, Cayley's Theorem, Sylow Theorems. Symmetric and Alternating groups, Solvable groups, Direct Products, Classification of Finite Abelian groups, Free groups, Group presentations. Rings, ideals, quotient rings, fields, Integral Domains, maximal and prime ideals, field of fractions, polynomial rings, Principal Ideal Domains, Euclidean Domains, Unique Factorization Domains, Gauss's Lemma, Eisenstein's Irreducibility Criterion, Chinese Remainder Theorem. Fields and field extensions. The Tower law. Algebraic and transcendental elements and extensions. Splitting field extensions. Algebraic closure. Normal and Separable extensions. Fundamental Theorem of Galois Theory. Finite fields. Cyclotomic extensions over Q. Solvability by radicals. Modules, direct sums, free modules and bases, torsion and torsion-free modules, finitely generated modules over a PID, tensor products (over commutative rings with 1), vector spaces, linear maps, dimension, matrices, minimal and characteristic polynomials, Cayley-Hamilton Theorem, Smith Normal Form, Rational Canonical Form, Jordan Normal Form.

Example Textbooks:

- Serge Lang,
*Algebra*3rd Ed.; - D.S. Dummit and R.M. Foote,
*Abstract Algebra*, 2nd Ed, Chapters 0-14; - L.C. Grove,
*Algebra*, Chapters I-IV; - N. Jacobson,
*Basic Algebra I, 2nd Ed.*Chapters 0-4.

Example exams: 2000F 2001F 2002S 2003S 2004S 2004F 2005S 2005F 2006F 2008S 2008F 2009F 2010F 2011S 2011F 2012S 2012F 2013S 2013F 2014S 2016F 2017S 2021 2022 2023

### Real Analysis, Math 7350-7351

Background topics: algebras and sigma-algebras of sets, Lebesgue measure and integration
on the real line, differentiation and integration, *L _{p}*-spaces, metric spaces, linear operators in Banach spaces, Hahn-Banach theorem, closed
graph theorem, general measure, signed measures, Radon-Nikodym theorem, product measure,
Fubini and Tonelli theorems.

Example textbooks:

- H.L. Royden,
*Real Analysis*, Macmillan Publishing Company 1988 (3rd edition). - H.L. Royden and P.M. Fitzpatrick,
*Real Analysis*, Prentice Hall 2010 (4th edition). - R.M. Duddley,
*Real Analysis and Probability*, Cambridge Studies in Advanced Mathematics 1989 (2nd edition). - S.K. Berberian,
*Fundamentals of Real Analysis*, Springer-Verlag 1999. - John N. McDonald and Neil A. Weiss,
*A Course in Real Analysis*, Academic Press 1999. - G.B. Folland,
*Real Analysis: Modern Techniques and their Applications*, Wiley-Interscience 1999.

Example exams: 2003S 2003U 2004S 2004F 2005S 2005F 2006S 2007F 2008F 2009F 2010S 2010F 2011S 2011F 2013S 2013F 2014S 2014F 2015S 2015F 2016S 2020F 2022 2023 2023F

## Ph.D. Qualifying Exam in Statistics

Students wishing to take the qualifying exams should consult with their advisors who will make the necessary arrangements with E. Olúṣẹ́gun George.

Example exams for Exam I:

2001 2002 2003 2004 2005 2006 2007 2009 2010 2011 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023

Example exams for Exam II:

2001 2002 2003 2004 2005 2006 2007 2009 2010 2011 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023

## Comprehensive Master's Exam – Non-Thesis Option

### Applied Mathematics/Mathematics Concentration:

The comprehensive master’s exam consists of a master’s presentation with a short, written summary on the topic of an independent study project, carried out under the guidance of a faculty member. Students are strongly advised to identify suitable faculty members early on. Students wishing to present a project should submit the title and a brief description of the project (one paragraph) to Prof. Gisèle Goldstein by September 30 in the fall semester and January 30 in the spring semester.

### Statistics Concentration:

The comprehensive master’s exam is a written exam covering the following topics: MATH 6636 - Intro Statistical Theory, MATH 7654 - Inference Theory plus two additional courses as described in the Graduate Catalog

### Teaching of Mathematics Concentration:

The comprehensive master’s exam is a written exam covering topics from four courses, each consisting of at least three credit hours of course work as described in the Graduate Catalog.

Students wishing to take the written comprehensive master’s exam in the Statistics or Teaching of Mathematics concentrations should consult with their advisors who will make the necessary arrangements with Prof. Pei-Kee Lin.

**Example exams** for the Statistics Concentration:

2001F 2002F 2003S 2003F 2004F 2005S 2005F 2006S 2006F 2007S 2007F 2008S 2008F 2009S 2009F 2010S 2010F 2011S 2011F 2012U 2013S

**Example Exams** for the Teaching of Mathematics Concentration:

2010S 2012S