Homework for Week 12

Part 1

  1. Prove that for every integer n the integers n2+1 and n3+2n are relatively prime.
  2. Prove that for every integer n the integer n14-n2 is divisible by 7.
  3. Find a polynomial f(x) in Q[x] of smallest possible degree such that
    • f(x) = (2x+1) mod (x2 + 1) and
    • f(x) = (x-1) mod (x2 +x+ 1).
  4. Find a polynomial f(x) in Q[x] of smallest possible degree such that
    • f(x) = x2 mod (x3 - 1) and
    • f(x) = (x+3) mod (x2 +2).
  5. Show that (2,x) is not a principal ideal in Z[x]. (This means Z[x] is not a PID.)
  6. (Rotman 3.78ii) Show that 2 and x are relatively prime in Z[x], but that 1 is not a linear combination of them, that is, there are no polynomials f(x), g(x) in Z[x] with 1 = 2f(x) + xg(x).
  7. (Rotman 3.79) Because x-1=( √ x  - 1)( √ x  + 1), a student claims that x-1 is not irreducible. Explain the student's mistake.

Part 2

  1. (Rotman 3.86 ii-iv, xi-xiv) True or false, with explanation.
    1. 13/78 is a rational root of 1 + 5x + 6x2.
    2. If f(x) = 3x4 + ax3 + bx2+cx+7 for some integers a,b,c then all rational roots (if any) of f(x) lie in the set {1, -1, 7, -7, 1/3, -1/3, 7/3, -7/3}.
    3. If f(x) = 3x4 + ax3 + bx2+cx+7 for some rational numbers a,b,c then all rational roots (if any) of f(x) lie in the set {1, -1, 7, -7, 1/3, -1/3, 7/3, -7/3}.
    4. If f(x)=g(x)h(x) in Q[x] and if f(x) has all its coefficients in Z then all the coefficients of g(x) and h(x) also lie in Z.
    5. For every integer c, the polynomial (x+c)2 - (x+c)-1 is irreducible in Q[x].
    6. For all integers n, the polynomial x8 + 5x3+5n is irreducible in Q[x].
    7. The polynomial x7+9x3+(9n+6) is irreducible in Q[x] for every integer n.
  2. (Rotman 3.87 iv-ix) Determine whether the following polynomials are irreducible in Q[x].
    1. f(x) = 8x3-6x-1
    2. f(x) = x3 + 6x2+5x+25
    3. f(x)=x5-4x+2
    4. f(x) = x4+x2+x+1
    5. f(x)=x4-10x2+1
    6. f(x)=x6-210x-616