Please explain this to me step by step Thank youSolution1a T

Please explain this to me step by step. Thank you.

Solution

1(a)

This is very easy. It just follows the addition law in Z₃. For example,

                         x+2 + x+1 =2x+3 =2x.

                        2x+ (x+2) = 3x+2 =0

                         (2x+1) +(x+2) = 3x+ 3 =0

Also note that addition is commutative. So the table is symmetric, enough to compute the elements above the diagonal.

1(b) Multiplication table for GF₃ [x]/(x² +x+1)

         Basic idea to multiply any two factors and use the fact x² +x+1=0 in the quotient.

Or replace x² by -x-1 = 2x+2.

For example,

                  (2x+1) (x+2) =2x² + 5x+2 = 2(2x+2) +5x+2= 9x+6=0..................(1)

                    (x+1)(x+2) = x² +3x+2 = 2x+2+2 = 2x+1

Again, you need to multiply only the pairs above the diagonal , as multiplication is commutative.

(1c)THIS IS NOT A FIELD, as there are zero divisors, for example (2x+1) and (x+2) , as above in(1)

(2a),(2b) follow the same method as in (1), except that replace x² by -2x-2 = x+1

For example,

(2x+1) (x+2)= 2x² +5x+2= 2(x+1)+2x+2= x+2

(2c) This is a field as the polynomial f(x)= x² +2x+2 is irreducible over Z_3.={0,1,2}, as can be verified by direct substitution.

                  f(0) =2, f(1) = 2 and f(2) = 1 (over Z₃)

So the quotient is a field (This can also be seen from the multiplication table , as every element has a (unique) inverse)

(3) (a,b) As has been seen above, the quotient ring GF₃[x]/(x² +2x+2) is a field. Represent the elements

{0,1,2,x,x+1,x+2,2x,2x+1,2x+2} as {0,1,2,4,5,6,7,8} (for example 8=2.3+2 , so represents 2x+2) respectively .

The addition and multiplication tables are just as in (2) but for the change in notation

3(c)

Note that 3 corresponds to x.

So 3² =x² =x+1

3³= x(x+1) = 2x+1

SImilarly work out the other powers of 3, as given in the table.

Note that 3 is an element of the of the multiplicative group (with 8 elements) of order 8. So 3 is a generator of this group, as required to be verified.

(3d) The other 3 generators are 3^k , where k is relatively prime to 8--so k =3,5 and 7.

The tables are computed in a similar fashion.