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Proof:
 
Proof:
  
We claim that the ideal <x,2> subset Z[x] is not principal. Suppose, to the contrary, that <x,2> = <p(x)> for some p(x) in Z[x].  p(x) cannot be a constant polynomial, because this constant would have to be even, and then x is not in <p(x)>.  Furthermore, p(x) has a nonzero constant term, since otherwise 2 is not in <p(x)>.  
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<nowiki>We claim that the ideal <x,2> subset Z[x] is not principal. Suppose, to the contrary, that <x,2> = <p(x)> for some p(x) in Z[x].  p(x) cannot be a constant polynomial, because this constant would have to be even, and then x is not in <p(x)>.  Furthermore, p(x) has a nonzero constant term, since otherwise 2 is not in <p(x)>.  
 
   
 
   
Since x is in <p(x)>, there exists a polynomial q(x) in Z[x] such that x = q(x)p(x), and since the constant term of q(x)p(x) is zero, q(x) has degree at least 1. But this means x = q(x)p(x) has degree at least 2, a contradiction.
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Since x is in <p(x)>, there exists a polynomial q(x) in Z[x] such that x = q(x)p(x), and since the constant term of q(x)p(x) is zero, q(x) has degree at least 1. But this means x = q(x)p(x) has degree at least 2, a contradiction.</nowiki>

Revision as of 17:38, 5 November 2008

Proof:

We claim that the ideal <x,2> subset Z[x] is not principal. Suppose, to the contrary, that <x,2> = <p(x)> for some p(x) in Z[x]. p(x) cannot be a constant polynomial, because this constant would have to be even, and then x is not in <p(x)>. Furthermore, p(x) has a nonzero constant term, since otherwise 2 is not in <p(x)>. Since x is in <p(x)>, there exists a polynomial q(x) in Z[x] such that x = q(x)p(x), and since the constant term of q(x)p(x) is zero, q(x) has degree at least 1. But this means x = q(x)p(x) has degree at least 2, a contradiction.

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