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Mathematics 2215: Rings, fields and modules Homework exercise sheet 3 Due 2.50pm, Monday 15 November 2010 1. Let R be an integral domain. (a) Explain why R[x] is an integral domain. (b) Show that deg(f g) = deg(f ) + deg(g) for any non-zero f, g ∈ R[x]. (c) Show that Units(R[x]) = Units(R). [For this question, you should pretend that R ⊆ R[x] by identifying an element a ∈ R with the constant polynomial a ∈ R[x].] 2. Let R be a subring of C with 1 ∈ R, let ζ ∈ C and let R[ζ] = {a0 + a1 ζ + · · · + an ζ n : n ≥ 0 and a0 , . . . , an ∈ R}. (a) Show that the map θ : R[x] → R[ζ], θ(f ) = f (ζ) is a is an surjective homomorphism, and describe its kernel. (b) Deduce that R[ζ] is a subring of C and an integral domain. (c) Explain why R[ζ] is isomorphic to a quotient of R[x]. 3. Let R be an integral domain. (a) Show that if f ∈ R[x] is a monic polynomial of degree 1, meaning that f = a + x for some a ∈ R, then f is an irreducible element of R[x]. (b) Show that the converse fails, by giving an example of an integral domain R and a monic polynomial f ∈ R[x] with deg(f ) > 1 so that f is an irreducible element of R[x]. You should prove that f is irreducible. √ √ 4. (a) Show that Units(Z[ 2]) = {a + b 2 : a, b ∈ Z and |a2 − 2b2 | = 1}. √ (b) Show that there are infinitely many units in Z[ 2]. [Hint: find a unit u with u > 1, and consider powers of u.] 5. Consider the set R= nn m o : n, m ∈ Z, m is odd . (a) Show that R is a unital subring of Q, and hence an integral domain. (b) What are the units of R? (c) What are the irreducible elements of R? (d) Is R a unique factorisation domain?