ABSTRACT
Section I in a Nutshell This section examines the integers (Z), the integers modulo m (Zm), and polynomials with rational coefficients (Q[x]). These structures share many algebraic properties:
• Each has addition defined and the addition is commutative. • Each has multiplicative defined and the multiplication is commutative. • Each has an additive identity (0 for Z, [0] for Zm, and the zero polynomial for Q[x]). • Each has a multiplicative identity (1 for Z, [1] for Zm, and the polynomial 1 for Q[x]). • All elements have additive inverses, but not all elements have multiplicative inverses. Furthermore, Z and Q[x] have some notion of ‘size’. The size of an integer is given by
its absolute value, while the size of a polynomial is given by its degree. This notion of size along with their similar algebraic properties, allow us to prove a
series of parallel theorems for Z and Q[x]:
(Theorem 2.1) Division Theorem for Z: Let a, b ∈ Z with a 6= 0. Then there exist unique integers q and r with 0 ≤ r < |a| such that b = aq + r. (Theorem 4.2) Division Theorem for Q[x]: Let f, g ∈ Q[x] with f 6= 0. Then there are unique polynomials q and r with deg(r) < deg(f) such that g = fq + r.
