National University of Sciences and Technology
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MATH-801 Algebra
Campus SNS
Programs PG
Session Fall Semester 2016
Course Title Algebra
Course Code MATH-801
Credit Hours 3-0
Course Objectives Aims

This lecture course aims to introduce students to the basic algebraic structures such as groups, rings and fields.

Brief Decommention of the Course

This course studies abstract algebraic structures. It attempts to understand the process of mathematical abstraction, the formulation of algebraic axiom systems, and the development of an abstract theory from these axiom systems. This course will cover the topics groups, rings, fields, Euclidean domains, principal ideal domains, unique factorization domains, and field extensions. Contemporary mathematics makes extensive use of abstract algebra. Subject areas such as algebraic number theory, algebraic topology and algebraic geometry apply algebraic methods to other areas of mathematics.

Learning Outcomes

On successful completion of this course, students will be able to
a)    know what is a group, subgroup, factor group, normal and centre subgroups;
b)    find number of all possible abelian groups up to isomorphism for a  given order;
c)    know what is a p-group and a sylow p-subgroup;
d)    show whether a group is simple or not;
e)    know what is a ring, subring and ideal;
f)    know polynomial ring, reducible and irreducible polynomials over certain rings;
g)    know integral domains, Euclidean domains, principal ideal domains, unique factorization domains;
h)    construct a bigger ring from a given ring;
i)    know what is field and construct fields from domains;
j)    know field extensions, algebraic extensions, splitting fields, algebraic closures, separable and inseparable extensions.
Detail Content
  • Groups, Subgroups, Cyclic Groups, Normal Subgroup, Centre Subgroup of a Group, Dihedral Groups;
  • Quotient Groups, Group Homomorphisms, Isomorphism Theorems for Groups,
  • Permutation Group, Cayley’s and Lagrange’s theorems;
  • p-Groups, Sylow’s Theorems and its Applications, Direct Product of Groups, Free Groups;
  • Fundamental Theorem of Finitely Generated Abelian Groups and its Applications,
  • Simple Groups, Group Action;
  • Rings, Subrings, Ideals, Quotient Rings, Maximal and Prime Ideals;
  • Ring Homomorphisms, Isomorphism Theorems;
  • Quadratic Integer Rings, Polynomial Rings, Formal Power Series Rings;
  • Integral Domains, Euclidean Domains, Principal Ideal Domains, Unique Factorization Domains;
  • Polynomial Rings over Fields, Irreducibility Criteria, Eisenstein Criterion,  Rings of Fractions, Fields of Fractions, Integral extension of rings;
  • Introduction to Field Theory, Field Extensions, Algebraic Extensions;
  • Splitting Fields, Algebraic Closures, Separable and Inseparable Extensions.
Text/Ref Books Text books
  • David S. Dummit, Richard M. Foote, Abstract Algebra, Jhon Wiley & Sons. (referred as DF).
Reference books
  • J. B. Fraleigh, A first Course in Abstract Algebra, Addison Wesely. (referred as JB).
  • I. N. Herstein, Topics in Algebra, John Wiley and Sons. (referred as IN).
Time Schedule Fall Semester 2014
Faculty/Resource Person Dr. Muhammad Ishaq
PhD GCU Lahore
Discipline: Commutative Algebra