5.16. Let F be a field, and suppose that the polynomial X² + X + 1 is irreducible in F[X]. Let K = F[X]/(X² + X + 1)F[X] be the quotient ring, so we know from Theorem 5.26 that K is a field. We will put bars over polynomials to indicate that they represent elements of K. In other words, if we let I be the ideal I = (X² + X + 1)F[X], then X + 2 is shorthand for the coset (X + 2) + I. (a) Find a polynomial p(X) = F[X] of degree at most 1 satisfying p(X) = (X + 3) · (2X + 1). (b) Find a polynomial q(X) € F[X] of degree at most 1 satisfying q(X) · (X + 1) = 1. In other words, find a multiplicative inverse for X + 1 in the field K. (c) Find a polynomial r(X) € F[X] of degree at most 1 satisfying

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