Elementary concepts (homomorphism, subgroup, coset, normal subgroup), solvable groups, commutator subgroup, Sylow theorems, structure of finitely generated Abelian groups. Symmetric, alternating, dihedral, and general linear groups.
Commutative rings and ideals (principal, prime, maximal). Integral domains, Euclidean domains, principal ideal domains, polynomial rings, Eisenstein's irreduciblility criterion, Chinese remainder theorem. Structure of finitely generated modules over a prinicpal ideal domain.
Extensions: finite, algebraic, separable, inseparable, transcendental, splitting field of a polynomial, primitive element theorem, algebraic closure. Finite Galois extensions and the Galois correspondence between subgroups of the Galois group and subextensions. Solvable extensions and solving equations by radicals. Finite fields.
Dummit and Foote, Algebra