Homework for Week 13

Part 1

  1. (Rotman 3.77) Let R be a PID and suppose r is an irreducible element of R. If s is in R and r does not divide s, prove that s and r are relatively prime.
  2. (Rotman 3.90)
    1. Let a be a squarefree integer other than 1, -1. Prove that xn-a is irreducible in Q[x] for every positive integer n. This shows that there are irreducible polynomials in Q[x] of every degree.
    2. Let a be a squarefree integer other than 1, -1. Prove that n √ a  is irrational.

Part 2

  1. For all of the following, suppose that a,b,c are integers.
    1. Prove that if gcd(a,b)=1 and c | a then gcd(c,b)=1.
    2. Prove that if a|b and a|c then a | gcd(b,c).
    3. Suppose that gcd(a,b)=c. Explain why a/c and b/c are integers. Now prove that a/c and b/c are relatively prime.
  2. For all of the following, suppose that f(x), g(x), and h(x) are polynomials in F[x] for some field F.
    1. Prove that if gcd(f(x), g(x))=1 and h(x) | f(x) then gcd(h(x), g(x))=1.
    2. Prove that if f(x) | g(x) and f(x) | h(x) then f(x) | gcd(g(x), h(x)).
    3. Suppose that gcd(f(x), g(x))=h(x). Prove that f(x)/h(x) and g(x)/h(x) are both polynomials. Now prove that f(x)/h(x) and g(x)/h(x) are relatively prime.
  3. For all of the following, suppose that R is a commutative PID and a,b,c are elements of R.
    1. Prove that if gcd(a,b)=1 and c | a then gcd(c,b)=1.
    2. Prove that if a|b and a|c then a | gcd(b,c).
    3. Suppose that gcd(a,b)=c. Explain which ring elements I mean when I write a/c and b/c. Now prove that a/c and b/c are relatively prime.