Algebra Qualifier

2004

Syllabus



Group Theory (Hungerford Chapter I)



Basic defintions and properties

Subgroups

Cosets

Congruences

Homomorphisms

Normal subgroups

Quotient group

Homomorphism and Isomorphism theorems

Index

Finite index

Orders of elements and subgroups

Finite groups

Free groups



Ring Theory (Hungerford Chapter III)



Basic definitions and properties

Integral domains

Fields

Ideals (left, right, 2 sided)

Homomorphism and Isomorphism theorems

Commutative rings

Localization

Local rings

Polynomial rings

Polynomials over a field

Noetherian rings



Fields and Galois Theory (Hungerford Chapter V)



Algebraic element

Minimal polynomial

Algebraic extension

Finite extension

Correspondence between intermediate fields and subgroups

Galois extension

Fundamental theorem of Galois theory

Separable extension

Splitting field



Linear Algebra



Vector space

Linear Transformation

Definitions and basic properties

Linear independence

Subspace

Span

Basis

Dimension