let (note that if then too. so the set we defined is not empty.)
so for some coprime with thus by the above remark for any coprime with now let we have where
therefore: because both and are in but this will contradict the minimality of unless so
this proves that obviously if and only if
Note: so every ideal of is cyclic. this problem is a good introduction to "localization", which is a powerful tool in the theory of commutative rings.