Homework for Week 11

Part 1

  1. (Rotman 3.29 ii-iii) True or false, with explanation.
    1. If R is a domain then R[x] is a domain.
    2. Let Q denote the rational numbers. Q[x] is a field.
  2. (Rotman 3.30) Show that if R is a nonzero commutative ring, then R[x] is never a field.
  3. (Rotman 3.46) Let R be a commutative ring. Show that the function f: R[x] → R given by f(a0 + a1 x + a2x2 + ...) = a0 is a homomorphism. Describe ker f in terms of roots of polynomials. What does the first isomorphism theorem say in this context?
  4. Suppose that R is a commutative domain and a is an element of R. Prove that the map f: R[x] → R defined by f(p(x)) = p(a) is a homomorphism. Describe in words the kernel ker(f). What does the first isomorphism theorem say here?
  5. Suppose that f: C[x] → C is the homomorphism defined by f(p(x)) = p(1). (C is the complex numbers.) Describe in words the kernel ker(f). What does the first isomorphism theorem say here?

Part 2

  1. (Rotman 3.56i-iv) True or false with explanation.
    1. If a(x), b(x) are polynomials in (Z/5Z)[x] with b(x) nonzero, then there are polynomials c(x), d(x) in (Z/5Z)[x] with a(x)=b(x)c(x)+d(x), where either d(x)=0 or deg(d(x)) < deg(b(x)).
    2. If f(x), g(x) are polynomials in Z[x] with g(x) nonzero, then there are polynomials q(x), r(x) in Z[x] with f(x)=g(x)q(x)+r(x), where either r(x)=0 or deg(r(x)) < deg(g(x)).
    3. The gcd of 2x2+4x+2 and 4x2+12x+8 in Q[x] is 2x+2.
    4. If R is a domain then every unit in R[x] has degree 0.
  2. Find the gcd of x2 - x - 2 and x3-7x+6 in Q[x] and express the gcd as a linear combination of the two polynomials.
  3. (Rotman 3.58) Find the gcd of x2 - x - 2 and x3-7x+6 in (Z/5Z)[x] and express the gcd as a linear combination of the two polynomials.
  4. (Rotman 3.59) Let k be a field and let f(x) be a nonzero polynomial in k[x]. Suppose that a1, a2, a3, ..., am are distinct roots of f(x) in k[x]. Prove that f(x) = (x-a1)(x-a2)(x-a3)...(x-a1)g(x) for some g(x) in k[x].
  5. (Rotman 3.61) Let R be an arbitrary commutative ring. Suppose that f(x) is a polynomial in R[x] and a in R is a root of f(x), so f(a)=0. Prove that there is a factorization f(x)=(x-a)g(x) in R[x].