Mathematics 2215: Rings, fields and modules
Tutorial exercise sheet 3
1. Let R be an integral domain. It is not true that
Gcds(a, b) = Gcds(s, t) =⇒ CDivs(a, b) = CDivs(s, t)
for all a, b, s, t ∈ R. Why not?
[The implication ⇐ is true, which we mentioned in class today].
This question is harder than I thought. However, it is technically correct! See the handwritten solutions.
2. If c is a gcd of two elements a, b in an integral domain R, and c0 kc,
show that c0 is also a gcd of a and b.
3. The ring Q[x] is a PID by Gauss’ lemma, so the ideal hx2 + 21 , 3x5 − 4i
is equal to hf i for some f ∈ Q[x]. Find such an f .
4. Is the function d : Z[x]× → N0 , d(f ) = deg(f ) a Euclidean function
on Z[x]? Explain why your answer is correct, and also why it is not
sufficient to answer the question: “is Z[x] a Euclidean domain?”.
5. If F is a field f ∈ F [x] with f 6= 0, and α ∈ F , show that
(a) If f = (x − α)q + r where q, r ∈ F [x] then f (α) = r(α).
(b) If f (α) = 0 then x − αf , and if deg f > 1 as well, then f is not
irreducible.
6. Find all of the irreducible polynomials of degree 2 in Z3 [x].