Combinatorics

Start with the field Z₃=({0,1,2}, + mod 3, mod 3). (1) Show that x²+1 cannot be factored in a nontrivial way (into polynomials of degree less than 2 with coefficients in Z₃). (2) Let be the root of this polynomial. Construct a Galois field with 3² = 9 elements. (3) Do the following computations: a) ², b) (1+), c) (1+)³, d) ^-1.