embers of GF(qm) are all pseudopolynomials in X of degree less than m over GF(q), and addition and multiplication in GF(qm) is polynomial addition and multiplication mod P(x), where x is a root of P(z). In the method of construction just outlined, GF(q) is called the ground field and GF(qm) is termed the (algebraic) extension field of GF(q). The term "extension field of GF(q)" may be used to describe any larger field con- taining GF(q); the modifier "algebraic" refers to the method of construction Pseudonoise Generators by which the field GF(q) is augmented with the root of an algebraic expres- sion to construct the larger field. Since each finite field contains an integer field having a prime number of elements, and since any field containing the integer subfield can be con- structed by the method stated in the above theorem, a finite field GF(q) con- taining q elements cannot exist unless q is a power of a prime number p. The prime p is called the characteristic of the field GF(q) when q = pn. Hence, finite fields containing 6, 10, 12, 14, 15, 18, (etc.) elements do not exist. The existence of fields containing 2, 3, 5, 7, 11, 13, 17, (etc.) elements has already been demonstrated, since these are prime fields (integer fields mod p). The existence of fields containing 4, 8, 9, 16, 25, 27, 32 (etc.) elements can be demonstrated; this is equivalent to showing the existence of an irreducible polynomial of the appropriate degree over the appropriate integer fields. The number N1(d) of irreducible polynomials over GF(q) of degree d can be shown to be (5A.40) where the sum is over all integer divisors of the degree d and u() is the Möbius function: in-form k = product of j distinct primes (5A.41) k divisible by a square. It is easily verified that N1(d) is a positive integer for all degrees d. 5A.6. Primitive Elements The non-zero elements of a Galois field GF(q) form a multiplicative group of order q - 1; hence, the exponent e of the non-zero elements of GF(q) under multiplication must divide q - 1 (see Section 5A.1)