1. You may apply results from the lecture notes, but you should clearly state which results you use and when you apply them.

(a) Let I = hX 2 −Xi be the principal ideal of Q[X] generated by X 2 −X and

let J = hX −1i be the principal ideal of Q[X] generated by X −1.

Define θ : Q[X] → Q[X]/I by θ(f(X)) = [X f(X)]I .

(i) Show that θ is a homomorphism.

(ii) Show that X −1 ∈ kerθ.

(iii) Show that kerθ = J.

(iv) Show that there is a subring S of Q[X]/I such that Q[X]/J ∼= S.

(b) Let m(X) = X 2 +X +2¯ ∈ Z5[X]. Let f(X) = X 4 +2X 3 −5X 2 −X +19 ∈ Z[X] and let ¯f(X) ∈ Z5[X] be the polynomial obtained from f(X) by reduction modulo 5.

(i) Show that m(X) is irreducible in Z5[X].

(ii) Show that ¯f(X) = m(X) 2 .

(iii) By using the factorisation of ¯f(X) in (b)(ii), or otherwise, show that f(X) is irreducible in Z[X].