Algebra I and II
Algebra I, II – 60210, 60220
Groups. Cyclic groups, permutation groups symmetric and alternating groups, matrix groups. Subgroups, quotient groups, direct products, homomorphism theorems. Automorphisms, conjugacy. Cosets, Lagrange's theorem. Group actions, G-sets, Sylow theorems, free groups and presentations.
Rings. Polynomial rings, matrix rings. Ideals, quotient rings, homomorphism theorems. Prime and maximal ideals. Principal ideal domains, Euclidean domains, unique factorization domains. Localization of rings, field of fractions.
Finite dimensional vector spaces. Basis, dimension, linear transformations, matrices. Modules. Submodules, quotient modules, homomorphism theorems, structure of finitely generated modules over a principal ideal domain. Language of categories and functors. Direct sums and products, free, projective and injective modules. Duality, multilinear forms, determinants.
Canonical forms of matrices of linear transformations
Jordan, rational and primary rational canonical forms. Invariant factors and elementary divisors.
Fields, algebraic and transcendental field extensions, degree, transcendence degree, algebraic closure. Fundamental theorem of Galois theory, separability, normality. Finite fields. *Cyclotomic extensions, *cyclic extensions, *solvable and nilpotent groups, *Impossibility proofs (trisecting angles, etc), *solvability of polynomial equations by radicals
Tensor products, *algebras, *(the tensor, symmetric and exterior algebras)
Artinian and Noetherian rings and modules, Hilbert basis theorem. *Simple and semisimple module
Localization of modules. Integral ring extensions. *Localization and spec of a ring. *Nakayama's lemma, *Noether normalization lemma. *Nullstellensatz. *Affine algebraic varieties and commutative rings.
- T. Hungerford, Algebra, Graduate Texts in Mathematics 73.
- N. Jacobson, Basic algebra I, II. S. Lang, Algebra.