Groups, subgroups, homomorphisms, cyclic groups, Lagrange's theorem,
normal subgroups, quotient groups, isomorphism theorems.
Jordan Holder theorem, solvable groups, symmetric and alternating groups.
S_n is not solvable, A_n is simple for n \geq 5.
Groups acting on sets.
Action on itself by conjugation. Orbit formula and the class equation.
Sylow theorems. Classification of groups of small order.
Categorical notions. Objects, morphisms. Products and co-products.
Universal objects.
Free groups. Groups defined by generators and relators.
Abelian groups. Fundamental theorem of finitely generated Abelian groups.
Presentation by matrices. Smith normal form of integer matrices.
Direct and inverse limits of groups. Examples.
Rings and ideals. Maximal and prime ideals. Nilradical and radical of ideals.
Examples. Commutative rings. Chinese Remainder Theorem.
Spec of a commutative ring and the Zariski
topology.
Factorial rings.
Modules over commutative rings and over PID. Tensor products of modules.
Tensor, symmetric and the exterior algebras.
Polynomials. Factoriality of A[X]. Gauss's lemma.
Noetherian rings. Hilbert's theorem on Noetherian rings.
Fields. Algebraic extensions. Splitting fields. Normal and separable
extensions. The primitive element theorem.