ALGEBRA
This is a standard graduate course on fundamental topics in algebra which are important in
many areas of mathematics. It covers the material for the qualification exam in Algebra.
Topics include group theory, commutative algebra, Noetherian rings, modules, rudiments of
category theory, field theory and non-commutative algebras.
Upon successfully passing the exam, the student should be able to:
-understand the basic algebraic concepts from the group theory commutative algbera,
modul and ring theory, Galois and relate these concepts with other areas of mathematics in
which they are used
-be able to use bibliography on Algebra in Croatian and English by themselves
Contents of the course: 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).
References: S. Lang, Algebra, Addison-Wesley, Reading, 1993. (third edition)