Mathematics 2215: Rings, fields and modules
Tutorial exercise sheet 2
1. (a) If R is an integral domain, explain why the only associate of 0R
is 0R .
(b) If a and b are associates in R, and R is a subring of an integral
domain S, then a and b are associates in S. Why?
(c) Compute
the associates of 3 in each of the following rings: Z, Z[i],
√
Z[ −5], Q and R[x].
(d) What are the associates of 3 + x in the ring R[x]?
2. Find all gcds of the following polynomials in the given ring R[x], where
(a) f = 4(1 − x2 ), g = 2(1 − x)(3 + x), R[x] = Z[x]
(b) f = 4(1 − x2 ), g = 2(1 − x)(3 + x), R[x] = Q[x]
3. Which of the seven polynomials [k]7 + [1]7 x2 for k = 0, 1, 2, . . . , 6 are
irreducible in Z7 [x]?
4. Let R be an integral domain. Call an element a ∈ R prime if a is
non-zero, a is not a unit of R and for all b, c ∈ R we have
a|bc =⇒ a|b or a|c.
Show that a prime element of R is irreducible.
5. Let F be P
a field. A polynomial f ∈ F [x] is called monic if it is of the
form f = nj=0 aj xj where an = 1F .
(a) Prove that every non-zero polynomial in F [x] is the associate of a
unique monic polynomial.
(b) Compute the monic associate of 1 + 3x + 2x4 ∈ Z5 [x]. [Strictly
speaking we should write this polynomial as [1]5 + [3]5 x + [2]5 x4 ,
but this notation gets tedious pretty quickly].
(c) Deduce that {f ∈ F [x] : f = 0 or f is monic} is acomplete set of
equivalence class representatives for the relation on F [x].
(d) Does this all work if you allow F to be any integral domain?