## ABSTRACT ALGEBRA NOTES BY AIM ACADEMY

### ABSTRACT ALGEBRA FOR CSIR NET MATHEMATICS

### ABSTRACT ALGEBRA HANDWRITTEN NOTES

The tests are conducted twice in a year generally in the months of June and December. It is conducted both in English and the Vernacular Language. The Council of Scientific and Industrial Research - CSIR has assigned the responsibility of conducting CSIR-UGC NET in CBT mode to NTA for science subjects namely Life Sciences, Chemical Sciences, Physical Sciences Mathematical Sciences and Earth Sciences jointly with the UGC.

**Syllabus of Mathematics**

**Unit 1** **Analysis**: Elementary set theory, finite, countable and uncountable sets, Real number system as a complete ordered field, Archimedean property, supremum, infimum. Sequences and series, convergence, Bolzano Weierstrass theorem, Heine Borel theorem. Continuity, Differentiability, Mean value theorem, Sequences, and series. Functions of several variables, Metric spaces, compactness, connectedness. Normed Linear Spaces.

**Linear Algebra**: Vector spaces, algebra of linear transformations. Algebra of matrices, determinant of matrices, linear equations. Eigenvalues and eigenvectors, Cayley-Hamilton theorem. Matrix representation of linear transformations. Change of basis, canonical forms, diagonal forms, triangular forms, Jordan forms.Quadratic forms, reduction, and classification of quadratic forms

**Unit 2** **Complex Analysis**: Algebra of complex numbers, the complex plane, polynomials, power series, transcendental functions such as exponential, trigonometric, and hyperbolic functions. Analytic functions, Cauchy-Riemann equations. Contour integral, Cauchy’s theorem, Cauchy’s integral formula, Liouville’s theorem, Maximum modulus principle, Schwarz lemma, Open mapping theorem. Taylor series, Laurent series, calculus of residues. Conformal mappings, Mobius transformations.

**Algebra**: Permutations, combinations, Fundamental theorem of arithmetic, divisibility in Z, congruences, Chinese Remainder Theorem, Euler’s Ã˜- function, primitive roots, Cayley’s theorem, Sylow theorems. Rings, ideals, prime and maximal ideals, quotient rings, unique factorization domain, Polynomial rings, and irreducibility criteria. Fields, finite fields, field extensions, Galois Theory.

**Topology**: basis, dense sets, subspace and product topology, separation axioms, connectedness, and compactness.

**Unit 3** Ordinary Differential Equations (ODEs): Existence and uniqueness of solutions of initial value problems for first-order ordinary differential equations, singular solutions of first-order ODEs, the system of first-order ODEs.

Partial Differential Equations (PDEs): Lagrange and Charpit methods for solving first order PDEs, Cauchy problem for first-order PDEs. Classification of second-order PDEs, General solution of higher-order PDEs with constant coefficients, Method of separation of variables for Laplace, Heat and Wave equations

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