Mathematics and Statistics

## Syllabus for Algebra

Sets and mappings:
One-to-one, onto and bijective maps
Equivalence relations and equivalence classes

Groups:
Uniqueness of identities and inverses
The order of an element

Subgroups:
Cosets
Lagrange's Theorem
Normal subgroups
Quotient groups

Group homorphisms:
Kernels and images
Isomorphisms
The basic homomorphism theorem (see .pdf for details)

Permutations:
S_n and cycle decomposition
Transpositions and A_n

Rings:
Commutative rings
Rings with unit element
Fields
Integral domains

Ideals:
Left and right ideals
Two-sided ideals
Quotient rings

Ring homomorphisms:
Kernels and images
Isomorphisms
The basic homomorphism theorem (see .pdf for details)

Quotient rings and fields:
Criteria for R to be a field
Maximal ideals
Criteria for R/M to be a field

Polynomial rings k[x], k a field:
The division algorithm
Every ideal in k[x] is principal
Irreducible polynomials and unique factorization
Maximal ideals in k[x]

Definitions and statements of the basic theorems are important