Mathematics homework help

Letf(x) =x3−x−1 inQ[x] and letg(x) =x2+ 1.a. Show thatf(x) is irreducible inQ[x].b. Quote general theorems which guarantee that the principal idealA= (f(x)) is a maximal ideal inQ[x] and the factor ringF=Q[x]/Ais a field.c. Find polynomialsr(x) ands(x) such thatf(x)r(x) +g(x)s(x) = 1.Suggestion. Use the same idea as the Euclidean algorithm forZ, invoving divisors and remainders.d. Explain why the cosetg(x) +Ais a unit inFand find a coset that represents its multiplicativeinverse.1