Algebra I and II
Algebra I, II – 60210, 60220
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Groups
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
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. -
Modules
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
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
Tensor products, *algebras, *(the tensor, symmetric and exterior algebras) -
Chain conditions
Artinian and Noetherian rings and modules, Hilbert basis theorem. *Simple and semisimple module -
Commutative algebra
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.
References
- T. Hungerford, Algebra, Graduate Texts in Mathematics 73.
- N. Jacobson, Basic algebra I, II. S. Lang, Algebra.