nt of each element of G. Let km denote the order of gm Then km is by definition the smallest integer such that gmkm=e, (5A.6) or, under the isomorphism, the smallest integer km such that mkm is a mul- tiple of k. Thus, km can be determined directly from k and m as mkm =1cm(k,m) (5A.7) Pseudonoise Generators where 1cm( ) denotes the least common (integer) multiple of the integers in parentheses. Example 5A.3. The cyclic group of Example 5A.2 has six elements and has primitive element 2. Solving (5A.7) leads to the following table, and verifies the calculations of Example 5A.2. element exponent 5A.2. Rings and Fields In the real number system we have learned to think in terms of two funda- mental operations: addition and multiplication. The abstract description of this is a field. DEFINITION 5A.4. A field is a non-empty collection F of elements which has the following properties: 1. F is a commutative group under an operation, + (addition), with the iden- tity element under + denoted by 0. 2. Excluding 0, the remaining elements of F are a commutative group under a second operation, . (multiplication), with the identity element under denoted by 1. 3. Distributive law: If f1, f2 and f3 are in F, then f1.(f2+f3)=(f1.f2)+(f1f3) Additive and multiplicative inverses of a field element will be denoted by - f and f1, respectively. Many algebraic structures have all but one or two of the properties of a field. Two structures, the integers under addition and multiplication, and the polynomials under addition and multiplication, have all but one of these properties; neither has multiplicative inverses for its elements. These are examples of structures, with addition and multiplication operations, which are called commutative rings. A commutative ring is an Abelian group Appendix 5A. Finite Field Arithmetic under addition, is closed, associative, and commutative under multiplication, and satisfies the distributive law. Two properties of rings and fields are the following: Property R1. Let r be an element of a ring R with addit