• Size and characteristic of a finite field
Concerning the field F_{2}, 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 F_{2} 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 = p^{k} for some integer k > 0 . Moreover, for all prime numbers p and all integers k > 0 there exists a field of size q = p^{k}. Thus the first finite fields according to their size are
F_{2}, F_{3}, F_{4}, F_{5}, F_{7}, F_{8}, F_{9}, F_{11}, ......
Operations in F_{3}
= { 0, 1, 2 }





+  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 F_{4} = { 0, 1, u, v }





+  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 
Observe that if a field F_{q} 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• Characteristic of F(q)
( X + Y )^{2} = X^{2} + 2*X*Y + Y^{2} = X^{2} + Y^{2}
( X + Y )^{p} = X^{p} + Y^{p}