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November 24th, 2010, 06:01 AM   #1
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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<?<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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