Reference no: EM132267002
Rings fields and Galois Theory (Advanced) Assignment - Questions
Q1. (a) Provide an example for each of the following. In each case give a proof that your example satisfies the given conditions.
(i) An irreducible element in the ring Q[x].
(ii) A maximal ideal in Z[x].
(iii) A unit u ≠ 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. Show that the map γ : R → S x T with γ(x) = (α(x), β(x)) is a ring homomorphism.
(d) Show that if n = paqb with p, q > 1 distinct primes and a, b ≥ 1 integers then
Z/nZ ≅ (Z/paZ) x (Z/qbZ).
Q2. (a) Let a(x), b(x) ∈ Q[x] be the polynomials
a(x) = x6 - 2x5 - 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 by
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) You are given that up to isomorphism there are exactly 4 distinct unital rings with precisely 4 elements. Find them all.
Q3. (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 + ½ denotes the set of all numbers of the form n + ½ with n ∈ Z (the half-integers). You are given that R is a commutative unital subring of C (you do not need to prove this).
(i) Let N : R → [0, ∞) be given by N(z) = |z|2. Show that N(z) ∈ N for all z ∈ R.
(ii) Find all units of R.
(iii) Decompose 25/2 - ½√-11 into irreducible factors over R.
(iv) Show that R is a principal ideal domain.
Attachment:- Assignment Files.rar