MNIT Jaipur Syllabus computer science Abstract Algebra

 

MNIT Jaipur Syllabus computer science   Abstract Algebra 

 

Abstract Algebra 

Number Systems: Natural numbers. Counting. Cardinality of finite sets. Laws, Mathematical induction.

Prime numbers. Fundamental theorem of arithmetic. Well-ordering principle. Number bases. Modulo

arithmetic.  Greatest  Common  Divisor,  Euler’s  extended  algorithm,  Chinese  Remainder  Theorem,

Primality testing,  Integers. Laws of arithmetic. Integer powers and logarithms. Recurrence relations.

Number sieves.

Group Theory: Groups,  Semi groups  and  Monoids, Cyclic semi graphs and sub monoids, Subgroups

and cosets, Congruence relations on Semi  groups, Factor groups and homomorphisms, Morphisms

Normal sub groups. Structure of   cyclic groups, Permutation groups, dihedral groups, Sylow theorems,

abelian groups; solvable groups, Nilpotent groups; groups of small order, elementary applications    in

coding theory.

Rings: Rings, Subrings, Morphism of rings, ideal and quotient rings, Euclidean   domains, Commutative

rings; integral domains, noncommutative examples, Structure of Noncommutative Rings, Ideal Theoryof Commutative Rings

Field Theory: Integral domains and Fields, polynomial representation of binary number, Galois fields,

primitive roots, discrete logarithms, split search algorithm.

Modules: Sums and products; chain conditions, Composition series; tensor products.

Text/ References:

1. John Fraleigh. First Course in Abstract Algebra, Pearson Education.

2. Michael Artin. Algebra, Pearson Education.

3. John A. Beachy and William D. Blair. Abstract Algebra, Second Edition, Waveland Press.

4. John A. Beachy. Abstract Algebra II, Cambridge University Press, London Mathematical Society

Student Texts #47, 1999.

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