Let $R$ be a principal ideal domain containing distinct prime elements $p, q$. Let $n, m \geq 1$ be integers. Is there a simple proof for the claim that the elements $p^n$ and $q^m$ of $R$ are relatively prime, i.e. that $(p^n) + (q^m) = R$, or equivalently that the greatest common divisor of $p^n$ and $q^m$ is $1$?
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Do you know that PIDs are UFDs? – Bill Dubuque Jan 31 '17 at 03:25
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@BillDubuque Yes. – User7819 Jan 31 '17 at 03:29
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2Then $,(p^m,q^m) = (d),\Rightarrow, d,$ divides the coprimes $,p^n,q^m,$ so $,d,$ is a unit, by unique prime factorization. – Bill Dubuque Jan 31 '17 at 03:31
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1Most all of the common proofs are ln this thread will work here. If for some reason those those proofs don't work for you then please refine your question to say why, and we can reopen. – Bill Dubuque Jan 31 '17 at 03:42