Course description:

The course is meant to be a second level introduction to algebra for students who are already familiar with the notions of groups and rings. We shall discuss some rudimentary results about various algebraic structures like commutative rings, modules, groups, and their interplay.

Topics:

• Prime and maximal ideals in commutative rings, review on principal ideal domains.
• Ideals in multivariate polynomial rings, Hilbert Basis Theorem.
• Modules over a ring, free modules, direct sums. Simple algebraic eld extensions, degree of an extension, product formula for the degree.
• Group actions, automorphisms of eld extensions, Galois extensions, splitting eld.
• Theorem on Galois correspondence, solvability of equations by radicals.
• Finite fields, Frobenius automorphism.
• Solvable groups, Jordan-Holder Theorem, Sylows Theorem.
• Symmetric polynomials, the discriminant.
• Smith normal form of matrices, structure theorem for nitely generated modues over principal ideal domains, applications ( finite abelian groups, Jordan normal form of matrices).
• Integral ring extensions, generalized Cayley-Hamilton Theorem.