Course Archives: Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore

Course Archives Theoretical Statistics and Mathematics Unit

Course: Algebra II
Level: Postgraduate
Time: Currently not offered

Syllabus: Results on finite groups: permutation groups, simple groups, solvable groups, simplicity of A_n.
Topics like semi-direct product (if not covered in Algebra-I).
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.

Time permitting, additional topics may be selected from:

(i) Traces and norms, Hilbert theorem 90, Artin-Schrier theorem, Galois cohomology, Kummer extension.
(ii) Transcendental extensions; Luroths theorem.
(iii) Real fields: ordered fields, real closed fields, Sturms theorem, real zeros and homomor-phisms.
(iv) Integral extensions and the Nullstellensatz.

Suggested Texts :

1. D.S. Dummit and R.M. Foote, Abstract Algebra, John Wiley (Asian reprint 2003).

2. S. Lang, Algebra, GTM (211), Springer (Indian reprint 2004).

3. M. Nagata, Field theory, Marcel-Dekker (1977).

4. N.S. Gopalakrishnan, University Algebra, Wiley Eastern (1986).

5. N. Jacobson, Basic Algebra, W.H. Freeman and Co (1985).

6. G. Rotman, Galois theory, Springer (Indian reprint 2005).

7. TIFR pamphlet on Galois theory.

Evaluation: