My confusion about Gaussian Integers
For some reason I was under impression that Gaussian Integers (http://en.wikipedia.org/wiki/Gaussian_integer) were not uniquely factorable and was trying to come up with a counterexample 🙂 For some reason I was sure that they were not a Euclidean ring. Well they ARE a Euclidean ring and hence a factorial ring.
(A million years ago, and a smaller number of light years away, when/where I studied math, we called unique factorization domains “factorial rings”. We didn’t use the term “integral domain” at all. Number theory was just part of “algebra”. Such bourbakism 🙂
Anyway, the Wiki article is surprisingly good. I learned about some unsolved problems that I didn’t know before http://en.wikipedia.org/wiki/Gaussian_integer#Unsolved_problems.
But when it comes to math I always check Mathworld in addition to Wiki (or even before Wiki). http://mathworld.wolfram.com/GaussianInteger.html
And the Mathworld article revealed the cause of my confusion. Gaussian integers only form a Euclidean ring if you define the norm as the sum of squares, and not as the square root (the modulus). I was so dead set on using the distance as the norm, that I didn’t think that the square of the distance is still a norm, and that it’s integer, so it works as a Euclidean degree function.
It’s so funny that the Euclidean distance is not a good norm for the Euclidean algorithm 🙂
I’m curious: can a real valued norm work as a degree function for the Euclidean algorithm at all?
Now I want to program the Euclidean algorithm and factorization for the Gaussians. But I doubt I’m ever going to do that – too busy/lazy.