|Course Archives Theoretical Statistics and Mathematics Unit|
Course: Algebra I
Instructor: Manish Kumar
Time: Curently offered
| Syllabus |
Syllabus: 1. Commutative rings with unity: examples, ring homomorphisms, ideals, quotients, isomorphism theorems with applications to non-trivial examples. Prime and maximal ideals, Zorns Lemma and existence of maximal ideals. Product of rings, ideals in a finite product, Chinese Remainder Theorem. Prime and maximal ideals in a quotient ring and a finite product. Field of fractions of an integral domain. Irreducible and prime elements; PID and UFD.
2. Polynomial Ring: universal property; division algorithm; roots of polynomials. Gauss Theorem (R UFD implies R[X] UFD); irreducibility criteria. Symmetric polynomials: Newtons Theorem. Power Series.
3. Noetherian rings and modules, algebras, finitely generated algebras, Hilbert Basis Theorem. Tensor product of modules: definition, basic properties and elementary computations. Time permitting, introduction to projective modules.
4. Groups: Review of normal subgroups, quotient groups and homomorhism theorems. Group actions with examples, class equations and their applications, Sylows Theorems; groups and symmetry. Direct sum and free Abelian groups. Time permitting: composition series, exact sequences, direct product and semidirect product with examples.
Note: It is desirable that Item No. 1 of Algebra I is covered before Item No. 2 of Linear Algebra begins.
1. D.S. Dummit and R.M. Foote, Abstract Algebra, John Wiley (Asian reprint 2003).
2. N. Jacobson, Basic Algebra Vol. I, W.H. Freeman and Co (1985).
3. S. Lang, Algebra, GTM (211), Springer (Indian reprint 2004).
4. N.S. Gopalakrishnan, University Algebra, Wiley Eastern (1986).
5. N.S. Gopalakrishnan, Commutative Algebra (chapter 1), Oxonian Press (1984).
6. J.J. Rotman, An Introduction to the theory of groups, GTM (148), Springer- Verlag (2002).
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