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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, ...... }

We observe that the sum of two even integers is an even integer. Further the sum of an even and an odd integer is an odd one, while the sum of two odd integers is an even one. Concerning multiplication, we see that the product of two even integers is an even integer. Also the product of an even by an odd integer is an even one, while the product of two odd integers is an odd one. Now we represent an even integer by 0 and an odd one by 1 , so each integer is represented by its remainder when divided by two. Then we can resume the above rules in the following tables

 

Operations in F 2 = { 0, 1 }
Addition




Multiplication
+ 0 1 * 0 1
0 0 1 0 0 0
1 1 0 1 0 1

 

F2 is a field since every element has an opposite and every non-zero element has an inverse. Observe that the opposite of 1 is 1 , while the inverse of 1 is also 1 .

 

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