Syllabus for Algebra I, JRF

1. Group theory:

   Group theory, permutation groups, Cayley's theorem, Sylow theorems, simple groups, solvable groups, composition series, semi-direct products, direct and inverse limits.

2. Ring theory:

   Ring theory, modules, integral domains and fraction fields, polynomial rings, matrix rings, PID, UFD, localization, exact sequences, projective modules, tensor products of modules, structure theorem for modules over PID.

3. Linear algebra:

   Vector spaces, direct sums, tensor products; Linear transformations and Matrices; Determinants; Dioganalizability and Nilpotence; Rational and Jordan form.

4. Spectral theorem:

   Bilinear forms; Inner product spaces; unitary, self-adjoint, normal, and isometric transformations; Spectral theorem.

5. Fields:

   Field theory: Algebraic and Transcendental extensions; Finite fields, Wedderburn's theorem on finite division rings.