Course Archives Theoretical Statistics and Mathematics Unit | ||||||||
Course: Algebra II Instructor: Manish Kumar Room: G25 Level: Postgraduate Time: Currently offered |
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Syllabus Past Exams Syllabus: i) Review of normal subgroups, quotient, isomorphism theorems, Group actions with examples, class equations and their applications, Sylows Theorems; groups and symmetry. Direct sum and free Abelian groups. Composition series, exact sequences, direct product and semidirect product with examples. Results on finite groups: permutation groups, simple groups, solvable groups, simplicity of An. ii) Algebraic and transcendental extensions; algebraic closure; splitting fields and normal extensions; separable, inseparable and purely inseparable extensions; finite fields. Galois extensions and Galois groups, Fundamental theorem of Galois theory, cyclic extensions, solvability by radicals, constructibility of regular n-gons, cyclotomic extensions. iii) Time permitting: Topics from Trace and Norms, Hilbert Theorem 90, Artin- Schreier theorem, Transcendental extensions, Real fields. Suggested Texts : (a) J.J. Rotman, An Introduction to the theory of groups, GTM (148), Springer-Verlag (2002). (b) D.S. Dummit and R.M. Foote, Abstract Algebra, John Wiley (Asian reprint 2003). (c) S. Lang, Algebra, GTM (211), Springer (Indian reprint 2004). (d) N.S. Gopalakrishnan, University Algebra, Wiley Eastern (1986). (e) N. Jacobson, Basic Algebra, W.H. Freeman and Co (1985). (f) G. Rotman, Galois theory, Springer (Indian reprint 2005). (g) TIFR pamphlet on Galois theory. (h) Patrick Morandi, Field and Galois Theory, GTM(167) Springer-Verlag (1996). (i) M. Nagata, Field theory, Marcel-Dekker (1977). Evaluation:
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Midterm
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