# Ph.D. Degree Requirements

Ph.D. candidates must complete a minimum of *36 credits (12 courses)* of graduate mathematics at the 500 level (excluding Math 500, 510-519, 599, 600, seminars, and directed study). These may include graduate courses taken for the M.S. degree.

**Note**: Students with prior graduate work at another university must take at least 18 credits (6 courses) of these courses at the University of Idaho.

The Graduate Catalog requires a minimum of 78 credits beyond the Bachelor of Science degree; however, that number can include Math 500, 599, 600, seminars, and directed study, as well as 400 level math courses and some supporting courses from outside mathematics.

## Preliminary Examinations

Three preliminary exams must be passed. These cover:

*Algebra*(555 and 557)*Analysis*(535, and one of 531, 536)

and one exam from one of the following three areas:

*Topology*(521 and 528)*Combinatorics*(two of 575, 576, 579)*Differential Equations*(539, 540)*Functional Analysis*(571, 572)

Course descriptions can be found in the online course catalog.

Students are **strongly encouraged** to make their first attempt at the preliminary examinations by the end of their second year. All three preliminary exams **should typically** be passed no later than the end of the fourth year of graduate study (including years at other universities). The three exams need not all be passed at once. Students will have a total of 3 attempts to pass the three preliminary exams, no matter how many they attempt each time

Prelims will usually be given according to the following schedule:

- Beginning the second full week of classes of the fall semester.
- Beginning the third week prior to finals week of the spring semester.

Preliminary exams are at a significantly higher level than M.S. exams.

When there is a choice of courses covered on a preliminary exam, the student may choose which two courses the exam will cover. Note that course work in the listed courses is not generally adequate preparation for preliminary exams. Each exam is written and is 4.5 hours in length. The following list of texts indicates the coverage on the different exams. Students should be prepared for questions covering any topic in the given texts.

## Reading List for Preliminary Exams

Students are advised to consult the professors making the exams to confirm that the list is current.

### Algebra Exam

Groups and Fields I (555):

- Rotman, The Theory of Groups
- Hungerford, Algebra
- Dummit and Foote, Abstract Algebra
- Garling, A Course in Galois Theory

Rings (557):

- Dummit and Foote, Abstract Algebra
- Hungerford, Algebra

### Analysis Exam

Real Analysis (535):

- Wheeden and Zygmund, Measure and Integral
- Royden, Real Analysis

Probability (536):

- Durrett, Probability: Theory and Examples
- Billingsley, Probability and Measure
- Chung, A Course in Probability Theory

Complex Analysis (531):

- Conway, Functions of One Complex Variable

### Topology Exam

Topology (521 and 528):

- Christenson and Voxman, Aspects of Topology (2nd ed.)
- Munkres, Topology (2nd ed.)
- Boothby, An Introduction to Differentiable Manifolds and Riemannian

Geometry

### Combinatorics Exam

Graph Theory I (575):

- Bondy and Murty, Graph Theory with Applications
- Chartrand and Lesniak, Graphs and Digraphs
- West, Introduction to Graph Theory

Graph Theory II (576) - *Same reading list as for 575*

Combinatorics (579):

- Hall, Combinatorial Theory
- Bondy and Murty, Graph Theory with Applications
- See instructor; this course can vary a lot and not all material is in texts

### Differential Equations Exam

Ordinary Differential Equations (539):

- Perko, Differential Equations and Dynamical Systems
- Hofbauer and Sigmund, Evolutionary Games and Population Dynamics
- Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra

Partial Differential Equations (540):

- Evans, Partial Differential Equations
- McOwen, Partial Differential Equations

### Functional Analysis Exam

Functional Analysis I and II (571 and 572):

- Eidelman, Milman, & Tsolomitis, Functional Analysis--An introduction
- Rudin, Functional Analysis

## Dissertation

The dissertation should contain original research and constitute a significant contribution to knowledge in the student’s field of study. Acceptability of the dissertation is to be determined by the student’s major professor and Ph.D. committee.