Version franšaise
• Size and characteristic of a finite field

Concerning the field F2, there is something striking and contrary to ordinary mathematics with the formula 1 + 1 = 0. Indeed the same type of formula holds in each finite field. There is a number p such that 1 + 1 + ..... + 1= 0 where 1 is repeated p times. This number p is a prime integer and is called the characteristic of the field. For instance we say that F2 is a field of characteristic 2 . It can be shown that, if a field has characteristic p then its number of elements is equal to q = pk for some integer k > 0 . Moreover, for all prime numbers p and all integers k > 0 there exists a field of size q = pk. Thus the first finite fields according to their size are

F2, F3, F4, F5, F7, F8, F9, F11, ......

There is no finite field of size 6 or 10 for instance. Fields of size p are called prime fields and they are easily represented by the integers 0, 1, 2,...., p-1 in a similar way as we did above for the field F2. A field of size q = pk with k > 1 is more sophisticated, but it contains the prime field Fp. We give below tables for operations in F3 and F4.

 

Operations in F3 = { 0, 1, 2 }
Addition




Multiplication
+ 0 1 2 * 0 1 2
0 0 1 2 0 0 0 0
1 1 2 0 1 0 1 2
2 2 0 1 2 0 2 1

 

Operations in F4 = { 0, 1, u, v }
Addition




Multiplication
+ 0 1 u v * 0 1 u v
0 0 1 u v 0 0 0 0 0
1 1 0 v 0 1 0 1 u v
u u v 0 1 u 0 u v 1
v v u 1 0 v 0 v 1 u

 

• Characteristic of F(q)
Observe that if a field Fq has characteristic p then for every element x of this field we have p*x = 0. Consequently according to the way addition has been defined in the field F(q), we also have p*X = 0 for every element X of F(q). We say that F(q) is an infinite field of characteristic p. For instance F(2) has characteristic 2 and we have 2*X = 0 for all formal number X. This property has an important consequence : for all X and Y in F(2) we have

( X + Y )2 = X2 + 2*X*Y + Y2 = X2 + Y2

Similarly for all X and Y in F(q) we will have

( X + Y )p = Xp + Yp

This formula will be of great help for computations in F(q). As an illustration in F(2), if X = 1011 then it implies easily X2 = 1000101 .

 

Go to page : 0 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 -10