Formal numbers
over a finite field
• Finite fields
At the begining of the nineteenth century, particularly with the work of E. Galois, the notion of finite field appeared. By finite field we mean a field with a finite number of elements. Towards the middle of the twentieth century, an intensive study of these sets and particularly of sets of functions over a finite field was undertaken. The fields of formal numbers over a finite field are of special importance. We will denote by F_{ n } a finite field with n elements and by F(n) the field of formal numbers over F_{ n } . When we consider this field F(n) , the analogies with the field of real numbers already mentioned are even more striking. The coefficients of a formal number in F(n) take only a finite number n of values. We state a nice consequence of that : the sequence of the coefficients for a formal number is ultimately periodic if and only if this number is the quotient of two formal integers.
• The smallest field
The simplest finite field is the set F_{ 2 } containing only 0 and 1 . We can illustrate this by considering two classical subsets of the natural integers : the even and the odd integers.
E = { 0, 2, 4, 6, ....... } and O = { 1, 3, 5, 7, ...... }






+  0  1  *  0  1  
0  0  1  0  0  0  
1  1  0  1  0  1 
F_{2} is a field since every element has an opposite and every nonzero element has an inverse. Observe that the opposite of 1 is 1 , while the inverse of 1 is also 1 .