Syllabus (MATHEMATICS)

Course Type: MAJ-12

Semester: 7

Course Code: BMTMMAJ12T

Course Title: Abstract Algebra-II & Linear Algebra-II

(L-P-Tu): 5-0-1

Credit: 6

Practical/Theory: Theory

Course Objective: To be done.

Learning Outcome: Course Outcomes (CO): The whole course will have the following outcomes: At the end of the course the students will be able to CO1: get concepts of direct product of finite number of group and group action etc. CO2: deal with ideal and isomorphism of ring

Syllabus:

Unit -1: Abstract Algebra-II [Credit: 3, 45L]

• Direct product of a finite number of groups, Group Actions, Orbit, Stabilizer, Class Equation, Cauchy’s Theorem, Sylow Theorems.
• Solvable group, Prime and Maximal ideal.
• Polynomial rings over commutative rings, Irreducibility of polynomials, Division Algorithm, Principal ideal domain, Euclidean domain.
• Field Extensions: Field extension, finite extension, simple extension, algebraic and transcendental extension and their characterizations. Splitting field, algebraic closure and algebraically closed field. Separable and normal extensions. Construction with straightedge and compass, finite fields and their properties, Galois group, Galois theory, Solvability by radicals, insolvability of the general equations of degree five (or more) by radicals.

Unit -2: Linear Algebra-II [Credit: 2, 30L]

• Introduction to linear transformations, algebra of linear transformation, rank and nullity of a linear transformation, matrix representation of a linear transformation.
• Dual Spaces, Dual Basis, Double Dual, Transpose of a linear transformation.
• Eigen Value, Eigen Vector and Eigen space of linear operator, diagonalizability, Cayley-Hamilton theorem for linear operator (Statement only), minimal polynomial for linear operator.
• Bilinear Form, Real Quadratic Form involving three variables, Reduction to Normal Form (Statements of relevant theorems and applications).
• Inner product spaces and norms, orthogonal & orthonormal set, triangular inequality, Schwartz inequality, Parallelogram law, Gram-Schmidt orthogonalization process, orthogonal complements, Bessel’s inequality, the adjoint of a linear operator.

Reading References:

1. John B. Fraleigh, A First Course in Abstract Algebra, 7th Ed., Pearson, 2002.
2. M. Artin, Abstract Algebra, 2nd Ed., Pearson, 2011.
3. Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence, Linear Algebra, 4th Ed., Prentice- Hall of India Pvt. Ltd., New Delhi, 2004.
4. Joseph A. Gallian, Contemporary Abstract Algebra, 4th Ed., Narosa Publishing House, New Delhi, 1999.
5. Gilbert, Linear Algebra and its Applications, Thomson, 2007.
6. Kenneth Hoffman, Ray Alden Kunze, Linear Algebra, 2nd Ed., Prentice-Hall of India Pvt. Ltd., 1971.
7. D.S. Malik, John M. Mordeson and M.K. Sen, Fundamentals of Abstract Algebra, McGraw-Hill Education-Europe.
8. I.N. Herstein, Topics in Algebra, Wiley Eastern Limited, India, 1975.
9. S. K. Mapa, Higher Algebra (Abstract and Linear), Levant Books.
10. M.K. Sen, S. Ghosh, P Mukhopadhyay, S. Maity, Topics in Abstract Algebra, Universities Press.