Syllabus
You can find the official syllabus for the qualifying exam here. For quick reference, that information is also recreated below. This local version also has the advantage that (eventually) terms can be linked directly to relevant notes.
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
Changelog