I have a pretty urgent question concerning the calculation of the class group, so any help will be very much appreciated:)(adsbygoogle = window.adsbygoogle || []).push({});

I'd like to illustrate my question with an example:

Calculate the ideal class group of Q(√-17), giving a representative ideal for each ideal class and a description of the group law.

My attempt at the question gets me up to the part:

(2) = (p_2)^2, where p_2 = (2, √-17 + 1) and N(p_2) = 2

(3) = (p_3)(q_3) where p_3 = (3, √-17 + 1), q_3 = (3, √-17 - 1) and N(p_3) = 3, N(q_3) = 3

where I've shown the other possibilities not to be viable as their norms are too large.

which leaves me with the ideals ϴ, p_2, p_3, q_3 which I've shown not to be principal.

The step proceeding this confuses me, it just says after this "we have N(1-√-17) = 18 = 2×3^2. Thus the possible decompositions of (1-√-17) are p_2×(p_3)^2, p_2×(q_3)^2 and p_2×(p_3)×(q_3)."

QUESTION i) I don't understand where the (1-√-17) comes from?

Furthermore, he goes on to give the relations of the group as (P_2)^2 = ϴ, (p_3)^2~(q_3)^(-2)~p_2 and finally (p_3)^4~ϴ, thus q_3~(q_3)^(-1)~(p_3)^3.

QUESTION ii) how did these relations come about?

Thanks again! I would be very grateful for any help given for either or both parts of my question:) if I missed something out or was unclear about something, please say:)

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# Computing the class group

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