Department of Mathematics

Syllabus

**Math 01.341 - Modern Algebra II**

**CATALOG DESCRIPTION:**

Math 01.341 Modern Algebra
II
3 s.h.

(Prerequisite: Math 01.340 Modern Algebra I with a grade of C- or better)

This course extends the study begun in Modern Algebra I to a more detailed investigation of abstract algebraic structures. Included are Sylow theorems, rings and ideals, polynomial rings, ring and field extension and Galois theory.

**CONTENT:**

1.** Group Theory**

1.1 Normal subgroup

1.2 Fundamental Theorem of Groups (Quotient Group)

1.3 Group isomorphism theorems

1.4 Cauchy's Theorem for abelian groups

1.5 Group automorphisms

1.6 Cayley's Theorem

1.7 Permutation groups

1.8 Sylow Theorems

2. **Ring Theory**

2.1 Ideal and prime ideals

2.2 Euclidean rings and principle ideal rings

2.3 Polynomials and zeros of a polynomial

2.4 Polynomial factorization theorem

2.5 Irreducibility criteria

2.6 Systemetric functions

3.** Theory of Fields**

3.1 Subfields and prime fields

3.2 Field adjunction

3.3 Simple field extensions

3.4 Linear dependence over a skew field

3.5 Linear equations over a skew field

3.6 Algebraic field extensions

3.7 Normal extension fields

3.8 Roots of unity

3.9 Finite commutative fields (Galois fields)

3.10 Separable and inseparable extensions

3.11 Perfect and imperfect fields

**TEXTS:**

*Joseph Gallian, CONTEMPORARY ABSTRACT ALGEBRA, Houghton Mifflin, 6th Edition

Rotman, J.J. A FIRST COURSE IN ABSTRACT ALGEBRA, Prentice Hall 2000 (2nd ed)

Pinter, C.C. ABOOK OF ABSTRACT ALGEBRA, (2nd ed) McCraw Hill, 1990.

Herstein, I.N. TOPICS IN ALGEBRA, 2nd. Ed., Xerox, Mass., 1975.