# Mathematics - Doctor of Philosophy

Candidates for the Ph.D. degree in the Department of Mathematical Sciences must pass

• a qualifying examination,
• a basic comprehensive written examination,
• an advanced comprehensive examination with written and oral components, and
• a final oral doctoral thesis examination.

These are briefly described below. For more information, see the Graduate School requirements in this catalog, and the Mathematics Graduate Student handbook at www.math.nmsu.edu.

## Course Requirements

Prefix Title Credits
Core Requirements
MATH 581Abstract Algebra II: Fields, Rings and Modules3
MATH 593Measure and Integration3
MATH 594Real Analysis3
MATH 595Introduction to Functional Analysis3

In addition to the core requirements, each student must complete one of the following four options.

Prefix Title Credits
Option 1
MATH 541Topology I3
MATH 542Topology II3
MATH 582Module Theory and Homological Algebra3
MATH 583Introduction to Commutative Algebra and Algebraic Geometry3
MATH 551Mathematical Structures in Logic3
MATH 552Universal Algebra and Model Theory3
Option 2
MATH 571Partial Differential Equations I3
MATH 572Partial Differential Equations II3
Choose two from the following:6
Module Theory and Homological Algebra3
Introduction to Commutative Algebra and Algebraic Geometry3
Mathematical Structures in Logic3
Universal Algebra and Model Theory3
Option 3
STAT 562Foundations of Probability3
STAT 571Continuous Multivariate Analysis3
STAT 572Linear Models3
Choose two from the following:6
Module Theory and Homological Algebra3
Introduction to Commutative Algebra and Algebraic Geometry3
Mathematical Structures in Logic3
Universal Algebra and Model Theory3
Option 4
STAT 562Foundations of Probability3
STAT 581Advanced Theory of Statistics I3
STAT 582Advanced Theory of Statistics II3
Choose two from the following:6
Module Theory and Homological Algebra3
Introduction to Commutative Algebra and Algebraic Geometry3
Mathematical Structures in Logic3
Universal Algebra and Model Theory3

A full time graduate student is required to register each semester for at least two Math/Stat graduate courses above MATH 530 Special Topics/STAT 525 Statistics: Theory and Applications. With the exception of the final semester, of these two, only one may be MATH 600 Doctoral Research or MATH 700 Doctoral Dissertation. Moreover, these two courses must not include:

Prefix Title Credits
MATH 562History and Theories of Mathematics Education3
MATH 563Algebra with Connections3
MATH 564From Number to Algebra3
MATH 566Data Analysis with Applications3
MATH 567From Measurement to Geometry3
MATH 568Using Number Throughout the Curriculum3
MATH 569Geometry with Connections3
MATH 599Master's Thesis15
STAT 598Special Research Problems1-3

## Qualifying Examination

Every student admitted to the Ph.D. program must pass a written qualifying examination. Students who complete their mathematics Master's degree at NMSU may request that the Master's written examination also fulfill the Ph.D. qualifying examination requirement. In all other cases, the student takes a written Ph.D. qualifying exam. This exam is identical to the Master's written exam (based on the courses 525/527) and is administered and graded at the same time, but a higher passing score is required than for a Master's student. Ph.D. candidates need to pass the exam within their first year.

## Basic Comprehensive Examinations

Every Ph.D. student must pass a Basic Comprehensive Exam, either in Algebra or in Analysis. The Algebra exam is based on the courses MATH 526 Abstract Algebra I: Groups and Rings and MATH 581 Abstract Algebra II: Fields, Rings and Modules and the Analysis exam is based on the courses MATH 528 Introduction to Real Analysis II and MATH 593 Measure and Integration.

These exams are offered every August and January. A student choosing to take the exam in Algebra must pass the exam within fourteen months of completing Math 581; similarly, a student taking the exam in Analysis must pass the exam within fourteen months of completing MATH 593. Students failing to pass the exams within this period of time may be removed from the program. Students will normally not be given more than two attempts at any one exam.