Sathyabama Institute of Science and Technology M.Sc - Mathematics SMT5209 Number Theory and Linear Algebra Syllabus

[Arvind Kumar]

Sathyabama Institute of Science and Technology M.Sc - Mathematics SMT5209 Number Theory and Linear Algebra Syllabus

DEPARTMENT OF MATHEMATICS FACULTY OF SCIENCE AND HUMANITIES

SMT5209
NUMBER THEORY AND LINEAR ALGEBRA
L T P CREDITS TOTAL MARKS
3 1 0 4 100

UNIT I 12 Hours
Divisibility theory in the integers – the division algorithm, the greatest common divisor, the Euclidean algorithm, the
Diophantine equation ax + by = c. Primes and their distribution. The fundamental theorem of arithmetic. The sieve of
Eratosthenes. The theory of congruences. Basic properties of congruence. Binary and decimal representation of integers.
Linear congruences and Chinese remainder theorem.
(Sections 2.2, 2.3, 2.4, 2.5, 3.1, 3.2, 4.2, 4.3 & 4.4 of Text 1).

UNIT II 12 Hours
Fermat's little theorem and pseudoprimes Wilson's theorem. The sum and number of divisors. The greatest integer function.
Euler's phi-function. Euler's generalization of Fermat's theorem. Properties of the phi-function. (Sections 5.2, 5.3, 6.1, 6.3, 7.2,
7.3 and 7.4 of Text 1) (Theorems 7.6 and 7.7 only).

UNIT III 12 Hours
Rank of a matrix – Elementary transformation, reduction to normal form, row reduced echelon form. Computing the inverse of
a non singular matrix using elementary row transformation. (Section 4.1 to 4.13 of Text 2)

UNIT IV 12 Hours
System of linear homogeneous equations. Null space and nullity of matrix. Sylvester's law of nullity. Range of a matrix.
Systems of linear non homogeneous equations. Characteristic roots and characteristic vectors of a square matrix.

UNIT V 12 hours
Some fundamental theorem. Characteristic roots of Hermitian, Skew Hermitian and Unitary matrices. Characteristic equation
of a matrix Cayley-Hamilton theorem.(Sections 6.1 to 6.6 and 11.1 to 11.3 and 11.11).

TEXT BOOKS
1. David M. Burton : Elementary Number Theory, Sixth Edn., TMH.
2. Shanti Narayanan & Mittal : A Text Book of Matrices, Revised edn., S. Chand.

REFERENCE BOOKS
1. C.Y. Hsiung : Elementary Theory of Numbers. Allied Publishers.
2. Neville Robbins : Beginning Number Theory, Second Ed. Narosa.
3. George E. Andrews : Number Theory, HPC.
4. Kenneth Hoffman & Ray Kunze : Linear Algebra, Pearson Education.