Syllabus
Topics covered
The examination covers basic properties of fundamental algebraic structures. The following is a (not necessarily exhaustive) list of topics from which questions are drawn.
Linear algebra
Vector spaces
- subspaces
- bases
- dimension
- direct sums
Linear transformations and matrices
- matrix representation of a linear transformation
- invertibility
- similarity
- determinants
- eigenvalues and eigenvectors
- diagonalization
Group theory
Groups
- basic properties of groups
- subgroups
- cosets
- Lagrange’s Theorem
- normal subgroups
- quotient groups
- cyclic groups
- permutation groups
- simple groups (definition of; simplicity of
for ) - Cayley’s Theorem
- direct products
- Fundamental Theorem of Finite Abelian Groups
Homomorphisms
- kernel
- image
- isomorphisms
- isomorphism theorems
- automorphisms
Ring theory
Rings
- basic properties of rings
- subrings
- ideals
- quotient rings
- ring homomorphisms
- isomorphism theorems
- direct sums
Integral domains and polynomial rings
- units
- associates
- Principal Ideal Domains (PIDs)
- Euclidean domains
- Unique Factorization Domains (UFDs)
- irreducible
- prime
- division algorithm
- criteria for irreduciblity
Fields
- characteristic
- construction via quotient rings
- fields as vector spaces, polynomial ring over a field
References
The material is found in a large number of texts, and is approached in a rather uniform fashion. Some texts that have been recently used are:
Linear algebra
- Axler, Linear Algebra Done Right
- Friedberg, Insel and Spence, Linear Algebra
- Lang, Linear Algebra
Group and ring theory
- M. Artin, Algebra
- Dummitt and Foote, Abstract Algebra
- Fraleigh and Brand, A First Course in Abstract Algebra
- Gallian, Contemporary Abstract Algebra
- Herstein, Abstract Algebra
Related pages
Home for the Algebra Qual
Problem bank
Notation Key
Past exams