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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)=ab?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<?<6 So each ideal class of Z[?2] contains an ideal I with ?I??5 Possible prime factors of I must lie over 2Z, 3Z, 5Z (Why must they lie over?) Futhere more one can show from some results that: 2R = (?2)^2 3R must be prime 5R must be prime Then the auther concludes that every ideal is principal (But why is that?) EXAMPLE 2: This time we look at K = Q[?5], R = Z[?5]. Integral basis: {1, ?5}. ? = (1+?5)(1+?5) (again it would be nice if someone could show me why we get these embeddings) = 1+2?5+5 then 10<?<11; so every ideal class contains an ideal with index ?I??10 So we know that a prime factor must lie over: 2Z, 3Z, 5Z or 7Z 2R = (2,1+?5)^2 Then we check if (2,1+?5) is principal (Why do we check this and not (2,1+?5)^2? I think it is because (2,1+?5) properly contains (2,1+?5)^2. But what if we wanted to check if 3R = (3,1+?5)(3,1?5) is principal? Do we have to check each of them? ) We suppose that (2,1+?5) is principal, such that (?)=(2,1+?5) then 2 = ?(2,1+?5)? = ?(?)? = N(?) (Do we get 2 as the index since R/2R have order 4 and R/(2,1+?5) must have order 2 (since (2,1+?5) properly contains (2,1+?5)^2). And therefore R/(2,1+?5) must be a proper divisor of 4, which leaves us no choice but 2?) We can let ? = a + b?5 such that N(?)= a^2 + 5*b^2 But this can never be equal to 2 (since a and b are integers) => (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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