 determining if (z - a) divides Q(z). This operation can be iterated until the number of (z - a) factors of P(z) is determined. The student should observe that "strange" things can happen in the fac- torization of polynomials over rings: (a) The number of roots of a polyno- mial over a ring, which are in the ring, may possibly be greater than or less than the degree of the polynomial. (b) Factorizations of polynomials over rings are not unique. Example 5A.7. Consider the following polynomials over the ring of inte- gers 0, 1, , 7 under addition and multiplication modulo 8. (a) z2 + 7 has four roots in the ring, namely 1, 3, 5, 7. (b) z2 + 1 has no roots in the ring. (c) z2 has two roots in the ring, namely 0 and 4, each of multiplicity 2. (d) Furthermore with an excess of roots available, factorization is certainly not unique. =(z)(z)=(z+4)(z+4) (e) There can be more factors on one side of an equality than the degree of the other side: (4z + 2 ) (2z + 4 ) 4 4 These unusual properties disappear when we require that the polynomial be over a field. Polynomials over any field satisfy certain basic factorization properties. These properties are most easily described in terms of a special kind of poly- nomial. DEFINITION 5A.5. A polynomial of degree n in an indeterminate Z is said to be monic if the (non-zero) coefficient of Zn is 1. The reader can easily verify that if a polynomial has coefficients in a field, then the following are true: (a) Every polynomial over a finite field can be uniquely written as a monic polynomial times an element of the coefficient field. The product of two monic polynomials of degrees m and n is a monic polynomial of degree m + N. (c) If a monic polynomial is the product of two polynomials, then it is the product of two monic polynomials. Pseudonoise Generators (d) If F(z) and H(z) are two polynomials over a field, which have greatest common divisor 1, then F(z)H(z)J(z) implies F(z)JJ(z). (e) A monic polynomial over a field F can always be uniquely factored into a product of irreducible mo