Ideal class group is isomorphic to ... - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-20T07:33:05Z http://mathoverflow.net/feeds/question/116436 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/116436/ideal-class-group-is-isomorphic-to Ideal class group is isomorphic to ... p.s.h. 2012-12-15T09:53:25Z 2012-12-15T10:16:23Z <p>In usual, ideal class group is defined by fractional ideals in Dedekind domain. But I read the chapter 12 of GTM84 (Springer) that defines the ideal class group by "integral ideals" as follows.</p> <blockquote> <p><strong>Definition.</strong> Two Ideals I and J are equivalent if (a)I=(b)J for some nonzero a, b in number ring D. Then, the equivalence classes can be made into a group, which is called ideal class group.</p> </blockquote> <p>And the last sentence of the last paragraph on p.185 of this book is as follows which makes me unhappy.</p> <blockquote> <p>It is not difficult to show that the ideal class group of an algebraic number field is isomorphic to the quotient group of the group of fractional ideals by the subgroupof principal fractional ideals.</p> </blockquote> <p>I AM DIFFICULT those groups are isomorphic, and I wonder if they are isomorphic not only number ring, but also arbitary Dedekind domain.</p> <p>Help me, my Santa Claus! And Merry Christmath :)</p>