Polynomials f(x) & g(x) in Z[x] Relatively Prime in Q[x]

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Discussion Overview

The discussion centers on the relationship between polynomials f(x) and g(x) in Z[x] and their status as relatively prime in Q[x]. Participants explore the conditions under which the ideal generated by these polynomials in Z[x] contains an integer.

Discussion Character

  • Technical explanation

Main Points Raised

  • One participant proposes that showing f(x) and g(x) in Z[x] are relatively prime in Q[x] is equivalent to demonstrating that the ideal they generate in Z[x] contains an integer.
  • Another participant argues that if the ideal contains an integer, then f(x) and g(x) are coprime in Q[x], explaining that if they are coprime, there exist polynomials a(x) and b(x) in Q[x] such that a(x) f(x) + b(x) g(x) = 1, leading to the conclusion that the ideal contains an integer.
  • A later reply expresses appreciation for the clarity of the previous explanation and indicates confidence in addressing the other implication of the discussion.

Areas of Agreement / Disagreement

Participants appear to agree on the implications of the first direction of the argument, but the discussion does not resolve the second implication or whether it is straightforward.

Contextual Notes

The discussion does not clarify the specific assumptions or definitions that underpin the claims made, nor does it address any potential limitations in the reasoning presented.

esisk
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trying to show that polynomials f(x), g(x) in Z[x] are relatively prime in Q[x] iff the ideal they generate in Z[x] contains an integer.Thanks .Not homework
 
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Z[x] contains an integer => f(x), g(x) coprime in Q[z] is easy. If f(x) and g(x) are coprime in Q[x], then there exist a(x) and b(x) in Q[x]. such that a(x) f(x) + b(x) g(x) = 1. There is an integer n such that n a(x) and n b(x) have coefficients in Z (think about why this is). Then n a(x) f(x) + n b(x) g(x) = n, so the ideal generated by f and g in Z[x] contains an integer.
 
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This is quite Rochfor1, thank you. And, yes, I will be able to do the other implication.
 
Sorry, I meant "quite clear". Thanks
 

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