- #1
AcidRainLiTE
- 89
- 2
Homework Statement
Show that the ideal
(3, x^3 - x^2 + 2x -1) \text{ in } \mathbb{Z}[x]
is not principal. (The parentheses mean 'the ideal generated by the elements enclosed in parentheses')
2. The attempt at a solution
I came up with a solution (see attachment), it is just rather convoluted. I feel like I am missing a more informative proof. I would like to take away as much as I can from this problem, so I am wondering if someone else sees a more insightful way to prove it. A brief summary of my solution is (see attachment for details):
(1) If it (call it I) is a principal ideal, then the fact that it contains 3 implies
I = a \mathbb{Z}[x] \text{ for some } a \in \mathbb{Z}
(2) But the fact that it contains x^3 - x^2 + 2x -1 implies a = 1 so
I = \mathbb{Z}[x]
(3) But 1 \notin I and hence I \ne \mathbb{Z}[x]. Contradiction. So I is not a principal ideal.