Review for final exam.

1.

Definitions of groups, subgroups, cosets, homomorphisms, normal subgroups, quotient groups. Go over the isomorphism theorems and Jordan-Holder theorem. Elementary examples of groups: symmetric groups S_n, alternating groups A_n, cyclic groups, dihedral groups D_2n.

2.

What does it mean to say that a group acts on a set ? What are orbits, isotropy subgroups ? How does a group act on itself by conjugacy ? Can you count ? Go over the counting formulas: Burnside's lemma, formula for counting orbits, the class equation.

3.

What are p-groups, Sylow theorems ? Using Sylow theorems to classify groups of small order. You should know all groups upto order 15.

4.

What are free groups ? How are groups presented by generators and relations ? What is the word problem ?

5.

What is the fundamental theorem for finitely generated Abelian groups ? What are Betti numbers and torsion coefficients ? What does it mean to present such a group by an integer matrix ? Given two such matrices how do you decide whether they present the same group ? Can you compute the Smith normal form of an integer matrix ?

6.

Are you familiar with the following categorical notions ? Categories, objects, morphisms, universal objects, (co-)products, fibred (co-)products, pull-backs and push-forwards, covariant and contravariant functors.

7.

Go over the definitions of rings, ideals, ring homomorphisms and fields. The Chinese Remainder theorem. What are factorial rings and principal ideal domains ? Examples of non-factorial rings and non-principal rings. What are local rings and localization ?

8.

What are modules ? Modules over PID are very similar to Abelian groups (which are modules over Z). Linear algebra of modules using matrices.

9.

What are exact sequences ? What are free modules, projective modules? Different criteria for bein projective. Can you chase diagrams ? You should be familiar with the snake lemma and the five lemma.

10.

The Hom(M,.) and the tensor product functors. Exactness properties of Hom and Tensor product functors.

11.

What is the tensor algebra of a module ? What are the symmetric and alternating algebras ? What are the dimensions of these algebras for a finite dimensional vector space ? Determinant as a wedge product.

12.

Are you familiar with the basic properties of polynomial rings in one variable: unique factorization of k[X]. You should be familiar with the following. Polynomials over a factorial ring. Content and Gauss lemma. Factoriality of polynomial rings over factorial rings. Eisenstein's criterion for irreducibility.

13.

What are Noetherian rings ? Examples of Noetherian and non-Noetherian rings. Hilbert's theorem: the rings R[X] and R[[X]] are Noetherian if R is.

14.

You should know about algebraic extensions, degree of extensions. irreducible polynomials of elements in an algebraic extension. Existence and uniqueness of the algebraic closure.

15.

What are splitting fields and normal extensions ? Existence and uniqueness of splitting fields. Separability and separable extensions. The primitive element theorem. Finite fields. Classification of all finite fields and their automorphisms.

16.

Good luck!