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## 541. Modern Algebra.

Credit Hours:
3
Course Level:
500
Prerequisites:
MATH 341.
Catalog Description:
Concepts from set theory and the equivalence of the Axiom of Choice. Zorn‚s Lemma and the Well-Ordering Theorem; a study of the structure of groups, rings, fields, and vector spaces; elementary factorization theory; extensions of ring and fields; modules and ideals; and lattices.
Semester Offered:
This is the basic graduate course in algebra. Note that algebra is one of the areas of the M.S. Advanced Exams/Ph.D. entrance exams. Math 541 follows in the spring with a second semester, Math 641, and both courses are needed if you are taking the exam. Math 541 is a basic prerequisite for advanced courses in combinatorics, graph theory, and number theory. It is a popular course for first year M.S. students and for first year Ph.D. students who expect to take the Entrance Exam in that area.

## 543. Linear Algebra. II, S.

Credit Hours:
3
Course Level:
500
Prerequisites:
MATH 441.
Catalog Description:
Review of theory of groups and fields; linear vector spaces including the theory of duality; full linear group; bilinear and quadratic forms; and theory of isotropic and totally isotropic spaces.
Semester Offered:
An undergraduate course in linear algebra is the expected background (either Math 343 or Math 441). The course is an abstract treatment of vector spaces and linear transformations, with additional topics, such as inner product spaces, dual spaces, etc. chosen depending on the instructor and the text. This course is required for all M.S. students (with rare exception) and a minimum grade of ‘B’ is needed.

## 545. Number Theory 1.

Credit Hours:
3
Course Level:
500
Prerequisites:
MATH 155 or MATH 156.
Catalog Description:
Introduction to classical number theory covering such topics as divisibility, the Euclidean algorithm, Diophantine equations, congruences, primitive roots, quadratic residues, number-theoretic functions, distribution of primes, irrationals, and combinatorial methods. Special numbers such as those of Bernoulli, Euler, and Stirling.
Semester Offered:
This course is offered about every other year and provides a graduate-level introduction to Number Theory. There is usually a second semester of Number Theory, Math 645, offered in the spring.

## 551. Real Variables 1.

Credit Hours:
3
Course Level:
500
Prerequisites:
MATH 451.
Catalog Description:
A development of Lebesgue integral, function spaces and Banach spaces, differentiation, complex measures, the Lebesgue-Radon-Nikodym theorem.
Semester Offered:
This is the first semester of a basic graduate two-semester course (551/651) in real analysis. Real analysis is one of the areas of the M.S. Advanced Exams/Ph.D. entrance exams and the full-year sequence should be taken if preparing for the exam. It is a prerequisite for the doctoral sequence in functional analysis and other doctoral-level courses in analysis and applied mathematics. You should have a good background in advanced calculus (Math 451) before taking this class. The first semester is largely devoted to developing Lebesgue measure. Math 651, Real Variables II, is offered in the spring.

## 555. Complex Variables 1.

Credit Hours:
3
Course Level:
500
Prerequisites:
MATH 451.
Catalog Description:
Number systems, the complex plane and its geometry. Holomorphic functions, power series, elementary functions, complex integration, representation theorems, the calculus of residues, analytic continuation and analytic function, elliptic functions, Holomorphic functions of several complex variables.
Semester Offered: