Math 3613
Homework Problems from Chapter 4
§4.1
4.1.1. Perform the indicated operations in Z6[X] and simply your answer.
(a) (3x4+ 2x3−4x2+x−4) + (4x3+x2+ 4x+ 3)
(b) (x+ 1)3
4.1.2. Which of the following subsets of R[x] are subrings of R[x]? Justify your answer.
(a) S={All polynomials with constant term 0R}.
(b) S={Alll polynomials of degree 2 }.
(c) S={All polynomials of degree ≤k∈N, where 0 < k}.
(d) S={All polynomials in which odd powers of xhave zero coefficients}.
(e) S={All polynomials in which even powers of xhave zero coefficients}.
4.1.3. List all polynomials of degree 3 in Z2[x].
4.1.4. Let Fbe a field and let fbe a non-zero polynomial in F[x]. Show that fis a unit in F[x] if and only
if deg f= 0.
§4.2
4.2.1. If a, b ∈Fand a6=b, show that x+aand x+bare relatively prime in F[x].
4.2.2. Let f, g ∈F[x]. If f|gand g|f, show that f=cg for some non-zero c∈F.
(b) If fand gare monic and f|gand g|f, show that f=g.
4.2.3. Let f∈F[x] and assume f|gfor every nonconstant g∈F[x]. Show that fis a constant polynomial.
4.2.4. Let f , g ∈F[x], not both zero, and let d=GCD (f , g). If his a common divisor of fand gof highest
possible degree, then prove that h=cd for some nonzero c∈F.
4.2.5. If fis relatively prime to 0F, what can be said about f.
4.2.6. Let f, g, h ∈F[x], with fand grelatively prime. If f|hand g|h, prove that fg |h.
4.2.7. Let f, g, h ∈F[x], with fand grelatively prime. If h|f, prove that hand gare relatively prime.
4.2.8. Let f, g, h ∈F[x], with fand grelatively prime. Prove that the GCD of f h and gis the same as the
GCD of hand g.
§4.3
4.3.1 Prove that fand gare associates in F[x] if and only if f|gand g|f.
4.3.2 Prove that fis irreducible in F[x] if and only if its associates are irreducible.
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