Norms, class groups, more, …

Sorry to bombard you with my problems professor, but i was attempting problem sheet 6 in a bid to understand how to calculate class groups properly, and have no real problem with it up until the point where we start to deduce which of the prime ideals are principal. When i say prime ideals i hope i’m right in calling the curly p with subscript of a prime number that. If the norm of a general element is a prime we say that the prime ideal is maximal right? I was attempting question one for root 11 and the one step i seem to have difficulty with is when we calculate the other norms of (3+(11)^1/2) and above, i have a vague understanding of what we do where we assign the norms to the prime ideals depending on what they are. But this step in general seems to allude me whenever i attempt these questions. So if you could shed some light on this step or could just guide me to the theorems or lemmas that would help with this area that would be extremely helpful. Thanks

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Reply:

I’m sorry to say this so close to the exam, but some of your questions are a bit worrisome. For example, the question `If the norm of a general element is a prime we say that the prime ideal is maximal right?’. It’s hard to make out what you mean. A correct statement is that inside the ring O_K of integers inside an algebraic number field K, all non-zero prime ideals are maximal. This fact is actually a bit tricky: it follows from the fact that O_K/I is finite for any non-zero ideal I and that any finite integral domain is a field. I hope you’re not confused about the *definition* of prime and maximal ideals, which just come from general algebra.

Let me guess a bit at what the confusion might be. When factorizing an ideal I, very relevant are the prime factors of N(I). This is because if

I=P_1P_2\cdots P_k

then

N(I)=N(P_1)N(P_2)\cdots N(P_k).

We know in fact that each N(P_i) will be a prime power factor of N(I). This allows us to look for prime ideal factors of I.

I hope you’ve already thoroughly reviewed the online notes. Chapter 4 is the relevant part for this material.

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