# MA8551 ALGEBRA AND NUMBER THEORY NOTES 2017 REGULATION

In this post we have posted some notes on MA8551 ALGEBRA AND NUMBER THEORY of BASIC SCIENCES for ANNA UNIVERSITY AFFILIATED COLLEGES STUDENTS.
Here We have listed the the notes whichever we could collect for MA8551 ALGEBRA AND NUMBER THEORY subject, We think that it will be helpful for your exam preparation.

#### OBJECTIVES:

ðŸ‘‰ To introduce the basic notions of groups, rings, fields which will then be used to solve related problems.
ðŸ‘‰ To introduce and apply the concepts of rings, finite fields and polynomials.
ðŸ‘‰ To understand the basic concepts in number theory
ðŸ‘‰ To examine the key questions in the Theory of Numbers.
ðŸ‘‰ To give an integrated approach to number theory and abstract algebra, and provide a firm basis for further reading and study in the subject.

#### UNIT I GROUPS AND RINGS

Groups : Definition – Properties – Homomorphism – Isomorphism – Cyclic groups – Cosets – Lagrange’s theorem. Rings: Definition – Sub rings – Integral domain – Field – Integer modulo n – Ring homomorphism.

#### UNIT II FINITE FIELDS AND POLYNOMIALS

Rings – Polynomial rings – Irreducible polynomials over finite fields – Factorization of polynomials
over finite fields.

#### UNIT III DIVISIBILITY THEORY AND CANONICAL DECOMPOSITIONS

Division algorithm â€“ Base – b representations â€“ Number patterns â€“ Prime and composite numbers â€“ GCD â€“ Euclidean algorithm â€“ Fundamental theorem of arithmetic â€“ LCM.

#### UNIT IV DIOPHANTINE EQUATIONS AND CONGRUENCES

Linear Diophantine equations â€“ Congruenceâ€˜s â€“ Linear Congruenceâ€˜s – Applications: Divisibility tests – Modular exponentiation-Chinese remainder theorem â€“ 2 x 2 linear systems.

#### UNIT V CLASSICAL THEOREMS AND MULTIPLICATIVE FUNCTIONS

Wilsonâ€˜s theorem â€“ Fermatâ€˜s little theorem â€“ Eulerâ€˜s theorem â€“ Eulerâ€˜s Phi functions â€“ Tau and Sigma functions.

#### OUTCOMES:

Upon successful completion of the course, students should be able to:
ðŸ‘‰ Apply the basic notions of groups, rings, fields which will then be used to solve related problems.
ðŸ‘‰ Explain the fundamental concepts of advanced algebra and their role in modern
mathematics and applied contexts.
ðŸ‘‰ Demonstrate accurate and efficient use of advanced algebraic techniques.
ðŸ‘‰ Demonstrate their mastery by solving non – trivial problems related to the concepts, and by proving simple theorems about the, statements proven by the text.
ðŸ‘‰ Apply integrated approach to number theory and abstract algebra, and provide a firm basis for further reading and study in the subject.

#### TEXTBOOKS:

1. Grimaldi, R.P and Ramana, B.V., “Discrete and Combinatorial Mathematics”, Pearson
Education, 5th Edition, New Delhi, 2007.

2. Koshy, T., â€•Elementary Number Theory with Applicationsâ€–, Elsevier Publications,
New Delhi, 2002.

#### REFERENCES:

1. Lidl, R. and Pitz, G, “Applied Abstract Algebra”, Springer Verlag, New Delhi, 2nd Edition, 2006.
2. Niven, I., Zuckerman.H.S., and Montgomery, H.L., â€•An Introduction to Theory of Numbersâ€–, John Wiley and Sons , Singapore, 2004.
3. San Ling and Chaoping Xing, â€•Coding Theory â€“ A first Courseâ€–, Cambridge Publications, Cambridge, 2004.

### MA8551 ALGEBRA AND NUMBER THEORY NOTES

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