AIMS OF THE PROPOSED COURSE: Study of general algebraic structures, groups rings, modules, algebras, fields and linear algebras.
Groups; basic notions and results. Categories and functors (definitions and basic examples, products and coproducts). Rings; basic notions, examples and results (homomorphisms, commutative rings, factorial rings). Modules (basic definitions, the group of homomorphisms , direct products and sums of modules, abelian categories). Localizations of rings and modules. Free modules. Modules over principal ideal rings. Polynomials (Euclidean algorithm, polynomials over unique factorization rings, reducibility and irreducibility, symmetric polynomials,rings of formal series). Noetherian rings and modules (basic results, noetherity of polynomial rings and rings of formal series, associated prime ideals, primary decomposition, Nakayama's lemma, filtered and graded rings and modules, Hilbert's polynomial). Algebraic spaces (Hilbert's Nullstellensatz, affine and projective varieties, spectrum of a ring). Algebraic field extensions (finite and algebraic extensions, algebraic closure, splitting fields and normal extensions, separable and inseparable extensions, finite fields). Galois theory (finite Galois extensions, roots of unity and cyclotomic fields , norm and trace, cyclic extensions, solvable extensions, infinite Galois extensions). Extensions of rings (integral extensions, integral Galois extensions, extensions of homomorphisms). Transcendental extensions (transcendence basis, Noetherian normalization theorem, linearly disjoint extensions). Matrices and linear mappings. Representations of an endomorphism (representations of algebras, decompositions induced by an endomorphism, characteristic polynomial). Bilinear forms (orthogonal summs, symmetric forms, orthogonal basis, Hermitian forms, the spectral theorems, alternating forms). Multilinear products (tensor product of modules, flat moduls, extension of the base, some functorial isomorphisms, tensor products of algebras, the tensor algebra of a module, symmetric and outer algebra). Semisimple modules (characterization of semisimple modules, Jacobson's density theorem, semisimple rings).