(Mathematics) ALGEBRA AND NUMBER THEORY B.Sc 2nd KUK/CRS University

Unit

Topics

Contact Hours

I

Symmetric, Skew symmetric, Hermitian and skew Hermitian matrices, Elementary operations on matrices, Rank of a matrix, Inverse of a matrix, Linear dependence and independence of rows and columns of matrix, Row rank and column rank of a matrix, Eigen values, Eigen vectors and characteristic equation of a matrix, Minimal polynomial of a matrix, Cayley-Hamilton theorem and its use in finding the inverse of a matrix, Unitary and orthogonal matrices.

12

II

Relations between the roots and coefficients of general polynomial equation in one variable, Solutions of polynomial equations having conditions on roots, Common roots and multiple roots, Transformation of equations, Nature of the roots of an equation, Descarte’s rule of signs.

12

III

Solutions of cubic equations (Cardon’s method), Biquadratic equations and their solutions. Divisibility, Greatest common divisor (gcd), Least common multiple (lcm), Prime numbers, Fundamental theorem of arithmetic.

12

IV

Linear congruences, Fermat’s theorem, Euler’s theorem, Wilson’s theorem and its converse, Chinese Remainder theorem, Linear Diophantine equations in two variables.

12