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**Algebra I, II – 60210, 60220**

Fall 2014 course web page

Spring 2015 course web page

**1. 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.

**2. 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.

**3. 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.

**4. ****Canonical forms of matrices of linear transformations**

Jordan, rational and primary rational canonical forms. Invariant factors and elementary divisors.

**5. 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

**6. Tensor products**

Tensor products, *algebras, *(the tensor, symmetric and exterior algebras)

**7. Chain conditions**

Artinian and Noetherian rings and modules, Hilbert basis theorem. *Simple and semisimple module

**8. 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*.