Hi,

I have a couple of questions concerning algebraic number theory. Why does contains imply divides ? Firstly assuming contains how do we know there exists a fractional ideal such that and then how do we get that is actually an integral ideal?

Another problem I am having is in proving: if is ideal and some prime then is prime ideal. So we assume are ideals such that is contained in . We want to show either is contained in or is. Assuming the above then we know divides . The proof then states that some ideal . I thought divides means for some ideal not other way round. I get that we could multiply by inverse to get on its own but then inverse is a fractional ideal.

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Reply:

I’m presuming that you’ve read the course notes at least twice or so at this point. So for your first question, it’s already assumed that we’ve shown that the non-zero fractional ideals form a group. This gives the existence of the you mention. But then

, so each element of stabilizes the ideal . This implies that every element of is an algebraic integer, i.e., that is an ideal.

For the statement about prime ideals, note that we are assuming the factorization theorem. So any can be written

for prime ideals . But then

Therefore, if is prime, only one can occur.