Show ring ideal is not principal ideal

  • #1

Homework Statement


Show that the ideal
[tex] (3, x^3 - x^2 + 2x -1) \text{ in } \mathbb{Z}[x] [/tex]
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
[tex] I = a \mathbb{Z}[x] \text{ for some } a \in \mathbb{Z} [/tex]

(2) But the fact that it contains [tex]x^3 - x^2 + 2x -1[/tex] implies a = 1 so
[tex] I = \mathbb{Z}[x] [/tex]

(3) But [itex] 1 \notin I [/itex] and hence [itex] I \ne \mathbb{Z}[x] [/itex]. Contradiction. So I is not a principal ideal.
 

Answers and Replies

  • #2
Forgot attachment
 

Attachments

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