## More norms of ideals

Dear Professor Kim,

I am unsure of how to calculate the norm of in the ring , which is on Sheet 5 Question 4a.

I can see that this ideal can be written as so it will have norm 4. Also, in the ring a general element looks like where belong to $ latex Z$. So if we calculate the norm using the principle

From what I understand this is the reasoning you give in ‘Some remarks on factorization’.

However, if we use the method which given further down that sheet I get:

Please tell me where I’m going wrong.

Many Thanks!

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Reply:

first of all, I hope you can see that the line

above doesn’t make too much sense. The second displayed equation is almost right, except an error occurs when computing

Because the coefficients are in , we have . So

from the isomorphism that takes to . It is easy to see that the -vector space has dimension 2 with basis . Hence,

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