## Factoring in algebraic number fields

Hi,

I was wondering if you could explain how to factorise ideals into prime and maximal ideals.

Also how do you calculate norms of ideals, as I have the definition N(I)=|Ok/I|, but I dont reall understand what this means.

Thanks for your help

Reply:

The main tool we’ve been using is Dedekind’s prime factorization theorem. Look at theorem 160 from the notes and examples 162 and 163 following it. There are a number of other examples in the optional coursework.

For a non-zero ideal I inside the ring O_K of algebraic integers inside a number field K, the notation

|O_K/I|

refers to the number of elements inside the quotient ring O_K/I.

If you would like to understand this, you can find computations in the examples mentioned above. You can even try easier ones:

-Take K=Q so that O_K=Z. What is the norm of the ideal (10)?

Inside Z[i], what is the norm of the ideal (2)? What about the ideal (1+i)?

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