In this exercise, we will construct a finite field f9 with 9 elements. consider the set f9 of all elements of the form a + bi, where a, b ? z3 and i is a fixed symbol.
(a) list all elements of f9.
remind yourself of how complex numbers are multiplied. now define multiplication of elements a + bi and a' + b'i in a similar way: (a + bi)(a' + b'i) = aa' ? bb' + (ab' + a'b)i, where aa' ? bb' and ab' + a'b are now executed in z3. as for complex numbers, this boils to using the rules for ordinary multiplication together with i^2 = ? 1.
(b) write down an addition table and a multiplication table for f9.
(c) use this to make a list of all elements of f9, together with their multiplicative inverse for each element of f9.