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I think everyone assume that it is very obvious, but actually I do not see why the following statement is obvious:

Class number 1 is equal that we have a UFD

So I tried to prove this for me:

$\rightarrow$ Class number 1 means every fractional ideal is a fractional principal ideal, so we have a PID and hence a UFD.

$\leftarrow$ We have a PID so a unique factorization into prime elements/ideals. And now I do not see how to go on.

Any suggestions? I hope it is not to obvious...

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    $\begingroup$ In a Dedekind domain, UFD and PID are equivalent. It can be proved that algebraic integers in a field is a Dedekind domain. You can consult any book on algebraic number theory for a proof. $\endgroup$
    – pisco
    Jan 16, 2018 at 7:11
  • $\begingroup$ Okay, that was easy, thanks :) $\endgroup$
    – Sqyuli
    Jan 16, 2018 at 7:17
  • $\begingroup$ Of course there are UFDs which are not Dedekind domains, and need not be PIDs. $\endgroup$ Jan 16, 2018 at 7:20

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