I’m not sure if this is what you are missing, but: Rings are closed, so you have to add an infinity of new numbers, not just the literal sqrt(5). a + b*sqrt(5)

I’m not sure if this is what you are missing, but: Rings are closed, so you have to add an infinity of new numbers, not just the literal sqrt(5). a + b*sqrt(5) is a convenient way to denote that all those numbers are in your system. So in some sense, once you’ve decided to incorporate sqrt(5), it really is only a ring once you add that part. Obviously there are tons of rings that have nothing to do with this recipe... maybe that is your nitpick?