suppose that F is a field with 3 elements01a prove that 11a

suppose that F is a field with 3 elements{0,1,a} prove that 1+1=a , a+1=0 ,a*a= 1.

let F is a field ,prove that if 1+1+1+1=0 in F ,then 1+1=0

Solution

Since F is a field , therefore first we construct the two composition table as for +3 and .3

See table no. 1

and See table no. 2

Now 1+1 =a because F is a field and F IS CLOSED WITH RESPECT TO THE ADDITION. (See table no. 1 )

ALSO a F , 1 F this implies that a+1=0 as F is closed with respect to the addition . (See table no. 1)

Further a F , again from the composition table a*a = 1 .(see table no. 2 )

since F is a field

also we suppose 1+1 = 0 then 1+1+1+1 = (1+1) +(1+1) =(0) + (0) = 0 from composition table 1 and using 1+1 =0.