Mathematics 2215: Rings, fields and modules
Homework exercise sheet 4
Due 2.50pm, Monday 29 November 2010
1. Find the gcds of 4x5 + x3 + 1 and 5x3 + 4 in Z7 [x].
2. Give a solution x, y ∈ Z to the equation 2010x + 1991y = 1.
3. Let R be an integral domain and let I C R. If a, b, q, r ∈ R with
a = bq + r and b ∈ I, show that I + a = I + r.
4. Let R be a Euclidean domain, with Euclidean function d : R× → N0 .
(a) If I C R and a ∈ I with a 6= 0, show that
I = hai ⇐⇒ d(a) = min{d(x) : x ∈ I, x 6= 0}.
(b) Deduce that if a, b ∈ R are both non-zero with a|b and d(a) = d(b),
then akb.
5. (a) Prove that Z[i] is a Euclidean domain, with Euclidean function
d : Z[i]× → N0 ,
d(z) = |z|2 .
Hint: to check the second condition in the definition of a Euclidean
function, draw a picture in the complex plane to explain why, if
a, b ∈ Z[i] with b 6= 0, you can choose q ∈ Z[i] so that | ab −q| ≤ √12 .
(b) Prove that if a, b ∈ Z[i]× and a|b in Z[i], then d(a)|d(b) in Z.
Then show that the converse of this statement is false, by finding
a, b ∈ Z[i]× so that d(a)|d(b) in Z but a6 | b in Z[i].
(c) Which of the following are irreducible elements of Z[i]?
2,
3,
4,
5,
1 + 2i
(d) Write 7 − i as a product of irreducible elements of Z[i].
(e) Find the gcds of 3 + i and 5 + i in Z[i].
(f) Prove that if I is a non-zero ideal of Z[i], then Z[i]/I is a finite
ring. [Hint: Exercise 3 might be useful.]