ｷ Revisiting the arithmetic and
algebra of the integers and the integers modulo *n*

ｷ Reviewing some basic ring theory
(ideals, quotient rings, fields)

ｷ Defining polynomials, polynomial
functions, and polynomial rings

ｷ Arithmetic in *k*[*x*], *k* a field (comparing it with the ring
of integers)

o Division algorithm

o Unique factorization (FT of Arithmetic)

o Irreducibility in Q[*x*], R[*x*] and C[*x*] (FT of Algebra)

o Structure of *k*[*x*]/(*p*(*x*)) (e.g., R[x]/(x2+1) and Z2[x]/(x2+x+1))

o Algebraic and transcendental extensions

o Elementary Galois
theory

ｷ Polynomial rings over commutative rings

o Division algorithm

o Unique factorization

ｷ Polynomial applications
(mathematical-Òreal worldÓ) appropriate for the secondary curriculum (this will
involve a review of the research literature)