Solution by radicals of equations of degree atmost 4, solvable groups, solvability by radicals, Abel-Ruffini theorem, symmetric functions, Newton's identities for symmetric functions, Galois groups of equations of degree at most 4. Equations with symmetric nd alternating groups as Galois groups. Reduction mod p technique. Cyclotomic extensions, norm and trace, cyclic extensions and Hilbert's theorem 90, Artin-Schreier theorem, transcendental extensions.
Zero divisors, nilpotent elements, nilradial and Jacobson radical, operations on ideals, extension and contraction, examples of rings arising in Geometry, Combinatorics, Number Theory and Topology.