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 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, ...... }
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0
1
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1
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1