Factoring in algebraic number fields


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


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


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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