nic polynomials over F. (f) Let P(z) be a polynomial of degree d over a field F, and let a1, be the distinct roots of P(z) in F with multiplicities m1 respec- tively. Then misd. (5A.24) If a polynomial P(z) over a field F can be written as a product of two or more polynomials (each with degree greater than 0) over the same field F, we say that P(z) is factorable over F On the other hand, if P(z) cannot be factored in this manner, P(z) is said to be irreducible over F. In specifying whether a polynomial is irreducible or factorable, the coefficient field be specified. Notice that the greateset common divisor of an irreducible poly- nomial F(z) and any other polynomial G(z) over the same field must be or F(z). Example 5A.8. z2 + 1 is irreducible over the real numbers. It is factorable over the complex numbers since z++1=(z+i)(z-i) where i is a name for a root of the polynomial z2 + 1 over the real numbers. 5A.4. Analyzing the Structure of Galois Fields A field with a finite number of elements q is called a Galois field, and is denoted by GF(q). The addition and multiplication tables of GF(16) are shown in Figure 5A.1, as are the tables for GF(2) (i.e., {0,1}) and GF(4) ({0, 1, a, b}). The reader may want to refer to GF(16) as a typical example of the abstract fields in the discussions to follow. Assuming that the addition and multiplication tables of a finite field GF(q) exist, let's explore the structures which GF(q) must contain. Suppose that the cyclic subgroup, I, generated by 1 under addition has p elements, I = {0, 1, p - 1}. (5A.25) I is the set of integers of GF(q), and it can be verified that addition and mul- tiplication of elements of I corresponds to arithmetic modulo p. It can be shown that subsets of an arbitrary field GF(q), which are closed under addition and multiplication and contain the identities 0 and 1, are in Appendix 5A. Finite Field Arithmetic fact fields in their own right. Hence, the following can be verified: THEOREM 5A.2. The integers 0, 1, 2, , p - 1, of a finite field form a sub- fiel