PAPER – I – ALGEBRA BHARATHIDASAN UNIVERSITY M.Sc. MATHEMATICS SYLLABUS (ANNUAL PATTERN) SYLLABUS

BHARATHIDASAN UNIVERSITY, TIRUCHIRAPPALLI – 620 024
M.Sc. MATHEMATICS SYLLABUS (ANNUAL PATTERN)
(FOR DISTANCE EDUCATION CANDIDATES ONLY)
(For the candidates admitted from the academic year 2006-2007 batch
onwards)

FIRST YEAR -- PAPER – I – ALGEBRA

Unit I
Another counting principle – Sylow’s theorems – Direct products – Finite abelian groups, Polynomial rings – Polynomials over the rational field – Polynomial rings over commutative rings.

Unit II
Extension fields – roots of polynomials – More about roots – The element of Galois theory – Finite fields – Wedderburn’s theorem on finite division rings – A theorem of Frobenius.

Unit III
The algebra of linear transformations – Isomorphism of vector spaces – Representations of linear transformations by matrices – Linear functionals- the double dual – the transpose of a linear transformation.

Unit IV
The algebra of polynomials – Lagrange Interpolation – Polynomial ideals – the prime factorization of a polynomial – Commutative rings – Determinant functions- Permutations and the uniqueness of determinant – classical adjoint of a matrix – Inverse of an invertible matrix using determinats.

Unit V
Characteristic values – Annihilating polynomial – Invariant subspaces –
Simultaneous triangulation – Simultaneous diagonalization – Direct sum decompositions.

Text books:

1. I.N. Herstein, “Topics in Algebra” Second Edition, Vikas Publishing House Pvt. Ltd., New Delhi.
Unit I: Chapter 2 ( 2.11,2.12,2.13,2.14) Chapter 3 (3.9,3.10,3.11)
Unit II: Chapter 5 (5.1,5.3,5.5,5.6) Chapter 7 (7.1,7.2,7.3)
2. K.Hoffman and R. Kunze, “ Linear Algebra”, Second Edition, Prentice – Hall of India Pvt. Ltd.
Unit III: Chapter 3 ( all sections )
Unit IV Chapter 4 (all sections ) Chapter 5 (5.1 to 5.4)
Unit V: Chapter 6 (6.1 to 6.6)
Books for References:
1. P.B. Bhattacharya, S.K. Jain, S.R. Nagpaul “Basic Abstract Algebra”, Cambridge University Press, Second Edition, 1995.
2. J.b. Fraleigh, “A First Course in Abstract Algebra”, Narosa Publishing House, New Delhi
3. N. Jacohson, Basic Algebra, Volume I and II, Freeman 1980.

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