Show that the polynomial f(x)-x in Z/6Z[x] factors as (3x 4)(4x +3), hence is not an irreducible polynomial. (a) Show that the reduction of f(x) modulo both of the nontrivial ideals (2) and (3) of Z./6Z is an irreducible polynomial, showing that the condition that R be an integral domain in Proposition 12 is necessary. (b) Show that in any factorization f(x)g(x)h(x) in Z/6Z[x] the reduction of g(x) modulo (2) is either 1 or x and the reduction of h(x) modulo (2) is then either x or 1, and similarly for the reductions modulo (3). Determine all the factorizations of f(x) in Z/6Z[x]. [Use the Chinese Remainder Theorem.] (c) Show that the ideal (3, x) is a principal ideal in Z/6ZIx].
