In a sense the task is large, but like a huge jigsaw puzzle. Once one gets a piece in place, it's there. Remembering is easier after one has forgotten it.
My confusion with the earlier link was that it talked about Z[sqrt(-3)], but that is an example of a ring that is not a Dedekind domain. It shows that they exist. One needs to start with Q, the rationals. Then note that Z is the ring of integers, OQ, in Q. An algebraic integer in an algebraic number field is the root of a monic polynomial with integer coefficients. In the case of Q, the monic polynomial is x - n = 0 where n is an integer. We get the expected ring of integers, Z.