can anyone help me with it?

8. Let f(x) = x4 - x - 1. Let F be a field such that FE {Q, R, C} U {Zp : p is prime}. Then f(x) E F[x], and so there are five options for the factorization of f(x) in F[x]: i. f(x) is irreducible in F[x]; ii. f(x) = g(x)(x - a), where g(x) is an irreducible polynomial of degree 3 and a E F; iii. f(x) = g(x)h(x), where g(x) and h(x) are irreducible polynomials of degree 2; iv. f(x) = g(x)(x - a1)(x - 02), where g(x) is an irreducible polynomial of degree 2 and a1, 02 E F; v. f(x) = (x -a1)(x -02)(x-03)(x-04), where a1, a2, 03, 04 E F. For example, in Z71 we obtain a factorization of type iii: ac4 - 2 - [1] = (22 + [15]ac + [51]) (x2 + [56]x + [32]), with x2 + [15]x + [51] and x2 + [56]x + [32] both irreducible in Z71 [x]. For the remaining types of factorizations (i, ii, iv and v) determine a field F such that f(x) factors in F[x] according to the specified type. Note: In this question you do not need to specify the factors of f (x) (they look pretty terrible in R and C). You are merely asked to prove that f (x) factors in a certain way