(Finite Fields) Let

n

be some fixed integer, and

a,b

some arbitrary integer. We say that\

a-=b,mod,n

\ if

a

and

b

satisfy\

a-b=kn

\ for some

kinZ

. For example,\

-3-=17mod,5

\ because\

-3-17=(-4)5.

\ Given a number

a

, which we call the representative, we denote by

a

the congruence class\ of

a

, so the set of all integers congruent to the representative. For example,\

[2]={dots,-8,-3,2,7,12,dots}.

\ Notice, we can always choose a representative satisfying

0

so that\

Z_(n)={[0],[1],dots,[n-1]}

\ is the set of all congruence classes. We can define addition and multiplication on this set by:\

[a]+[b]=[a+b]\ [a][b]=[ab]

\ a) Show that addition and multiplication as defined above is well defined. Notice, at the\ moment they are defined in terms of representatives. We need to show that the definitions\ do not depend on the representatives, so they only depend on the congruence class. This\ means that if\

a_(1)-=a_(2),mod,n\ b_(1)-=b_(2),mod,n

\ then\

[a_(1)]+[b_(1)]=[a_(2)]+[b_(2)]\ [a_(1)][b_(1)]=[a_(2)][b_(2)].

abmodn if a and b satisfy ab=kn for some kZ. For example, 317mod5 because 317=(4)5 Given a number a, which we call the representative, we denote by [a] the congruence class of a, so the set of all integers congruent to the representative. For example, [2]={,8,3,2,7,12,}. Notice, we can always choose a representative satisfying 0rn1 so that Zn={[0],[1],,[n1]} is the set of all congruence classes. We can define addition and multiplication on this set by: [a]+[b][a][b]=[a+b]=[ab] a) Show that addition and multiplication as defined above is well defined. Notice, at the moment they are defined in terms of representatives. We need to show that the definitions do not depend on the representatives, so they only depend on the congruence class. This means that if a1a2b1b2modmodnn then [a1]+[b1][a1][b1]=[a2]+[b2]=[a2][b2]