Lots of texts online on number theory do not even mention the different ideal. Some do, but then it gets ignored and is never mentioned again. I could not find a single application for it, as if it is just some random concept invented so teachers can make students do computations. I tried to search it, but the word "different ideal" is so common that I keep getting psychology webpages instead. Now, you can calculate the ramification from the difference - but normally to get the difference you need to know the ramification, so that is kind of pointless. I have heard that you can find the genus from the difference, but there is no genus in number field. So what is the point of this?
Discriminant seemed to help with figuring out what the ring of integer is. Is that all? Is there a good example of it being used when more elementary method failed?
Thank you.
I hope this text will be useful: http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/different.pdf