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Let $f,g$ be polynomial in $\Bbb{Z}[x]$ which are coprime, does this implies $f,g$ are also coprime in $\Bbb{Q}[x]$?

Similarily if $f,g$ are coprime in $\Bbb{C}[x,y]$ does this implies $f,g$ are coprime in $(\Bbb{C}(x))[y]$?

(I know some similar result that is if $f$ is premitive polynomial of positive degree then $f$ irreducible in $\Bbb{Z}[x]$ also irreducible in $\Bbb{Q}[x]$.)

As Magdiragdag point out

  1. if $f,g$ relative prime is defined to be that $f,g$ do not share a common factor, then the implication does not hold
  2. Another definition of being relative prime is given in stack project which is defined to be $(f)+(g) = (1)$, however it is only defined for the field(with two definitions coinside over field as we discussed in the comment.). If follow this definition the result holds trivially.
yi li
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1 Answers1

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Yeah. If $f, g$ are coprime in a ring $R$, that is $af+bg=1$ for some $a,b\in R$, then certainly it is still the case in any extension ring $R'\supset R$.

Just a user
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  • Not true for general rings. See https://en.wikipedia.org/wiki/B%C3%A9zout_domain – lhf Jan 04 '23 at 10:27
  • Please strive not to post more (dupe) answers to dupes of FAQs, cf. recent site policy announcement here. – Bill Dubuque Jan 04 '23 at 14:16
  • Hi @lhf , sorry for the silly question can you provide a concrete example that $f,g$ relative prime in $\Bbb{Z}[x]$ but not in $\Bbb{Q}[x]$? – yi li Jan 05 '23 at 13:16
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    $2$ and $x$ have no common factor but the ideal the generate is not the whole $\mathbb Z[x]$ – lhf Jan 05 '23 at 15:28
  • thank you @lhf , you showed that relative prime elements needs not to generate the whole $\Bbb{Z}[x]$. however this seems not to be the example that I ask (I'm not sure) since $2,x$ has no common factor in $\Bbb{Z}[x]$ but it also has no common factor in $\Bbb{Q}[x]$ – yi li Jan 06 '23 at 01:46