1

I'm having trouble proving that the ideal generated by $ \langle x,x+2 \rangle $ in the polynomial ring over the integers is not principal. I think I understand the problem I'm just not sure how to actually prove it. If we have two first degree polynomials in ring of the form $(x+a),(x+b)$ then their product obviously has the form $(x^2+(a+b)x+ab)$ which obviously is equivalent to $x^2+bx$ if one of the constants is zero. I think that's the main property that I should focus on but I'm not sure of how I could progress, if someone could help I'd appreciate it.

J. W. Tanner
  • 63,683
  • 4
  • 43
  • 88
Alp
  • 748
  • 1
    What you think is the main property is not the main property, since you're completely ignoring the role of 2. Write the ideal as $(2,x)$. It's the set of polynomials with an even constant term. – KCd Dec 11 '22 at 01:03
  • 1
    Note that $2$ is in the ideal. – Ted Shifrin Dec 11 '22 at 01:03

0 Answers0