There's a question that says, "Prove that the ring of integers O in the quadratic integer ring Q is a Euclidean Domain.
What do they mean by ring of integers O?
I know Q ={q + r ;q,r in Q}
... so would the ring of integers be O = {q + 0 ;q in Z} or do they mean algebraic integers or something?
Ohh okay, I think I can prove that. In another class I proved that Z[ ] was a Euclidean Domain using the same norm, so I'm guessing it'll be pretty similar. Does the fact that they mentioned Q[ ] change anything, or do I just procede the same as if they said Z[ ] right off the bat?
well, that is positive, but your "norm" doesn't correspond very well to the notion of "distance from 0". for example, your norm makes 1+√2 the same "distance" away from 0 as 1.
use drexel28's norm, i assure you, he is experienced in such matters.
Hmmm, okay, I'm just trying to understand where it came from. Because in all of the examples in class, we found the norm by multiplying x by its complement. And in my book they just say the norm in general is a^2 - Db^2. Here they seem to use the absolute value: The integers with the square root of 2 adjoined is a Euclidean domain « Project Crazy Project .
What you are referring to is sometimes called the ' -inequality' There is no need to show it. It's a cool fact though that any Euclidean domain admits a Euclidean function (function which allows the Division Algorithm to work) also admits a Euclidean function which satisfies the -inequality. It is usually given by .