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?