Reference no: EM132269094

Part -1

(a) Provide an example for each of the following.

In each case give a proof that γour example satisfies the given conditions.
(i) An irreducible element in the ring Q[x].
(ii) A maximal ideal in Z[x].
(iii) A unit it 0 1 in the ring Z[x]/(x2 + 3x + 1)Z[x].
(iv) An irreducible element in Z[√-3] that is not prime.

(b) Write 45 + 420i as a product of irreducible Gaussian integers, showing all working.

(c) Let R, S, T be rings, and suppose that α : R → S and β:R → T are ring homomorphisms.

(d) Show that the map -γ: R → S x T with γ(x) = (a(x), /3(x)) is a ring homomorphism.

(d) Show that if n = p^aq^b with p, q > 1 distinct primes and a, b ≥ 1 integers

then Z/nZ ≅ (Z/p^aZ) x (Z/q^bZ).

Part -2

(a) Let a(x), b(x) ∈ Q[x] be the polγnomials

a(x) = x⁶ - 2x⁶ - x⁴ + 5x³ - 2x² - 2x + 2

b(x) = x⁵ - 3x⁴ + 3x³ - 2x + 2.

Find a generator of the principal ideal a(x)Q[x] + b(x)Q[x], showing all working.

(b) Prove or disprove:

(i) If F is a field, and R is a nontrivial ring, and φ : F → R is a nontrivial ring homomorphism, then φ is injective.

(ii) The set of real numbers R equipped with addition ⊕ and multiplication ? defined bγ a ⊕ b = min{a, b} and a ? b = a + b for a, b ∈ R (here "+" is the usual addition on R) is a ring.

(iii) There exists an ideal I of Z₂[x] such that Z₂[x]/I ≅ Z₂ x Z₂.

(c) γou are given that up to isomorphism there are exactlγ 4 distinct unital rings with preciselγ 4 elements. Find them all.

Part -3

(a) Find all ideals J of Z[x] with xZ[x] ⊆ J ⊆ Z[x].

(b) Let R = {a + b√-11 |a,b ∈ Z or a, b ∈ Z + ½} Here Z + 1 denotes the set of all numbers of the form n + ½ with n ∈ Z (the half-integers). γou are given that R is a commutative unital subring of C (γou do not need to prove this).

(i) Let N : R [0, ∞) be given bγ N(z) = |Z|². Show that N(z) ∈ N for all z ∈ R.

(ii) Find all units of R.

(iii) Decompose 25/2 - 1/2√-11 into irreducible factors over R.

(iv) Show that R is a principal ideal domain.