My Math Forum example of Ideal class group

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 November 24th, 2010, 06:01 AM #1 Newbie   Joined: Nov 2010 Posts: 10 Thanks: 0 example of Ideal class group Hi I have some troubles understanding some notes about some examples in the ideal class. Here is a required theorem to solve the examples: Thm: For every numberring R there is a real consant ? such that every ideal I ? R contains a nonzero element ? with |N(?)|???I? EXAMPLE 1: K = Q[?2], R = Z[?2]. Integral basis: {1, ?2}. ?_1=id, ?_2=K?C, ?_2 (a+b?2)=a-b?2 (I am not yet very comfortable with embeddings, so it would be nice if someone could show me why we get these embeddings) ?_(i=1)^2?(?_(j=1)^2?|?_i (?_j)| ) =(1+?2)(1+?2)=1+2?2+1 (Here we just take [|?_1 (?_1)|+|?_1 (?_2)|]*[|?_2 (?_1)|+|?_2 (?_2)|], but I can`t quite see how this becomes (1+?2)(1+?2) ) which means that 5 (2,1+?-5) cannot be principal and therefor is an element of order 2 in the ideal class group... I would really appreciate it if someone could shed some light on either one or both of these examples. Thanks for all your time and effort.

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### ideal class group example

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