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Old 2019-07-03, 19:04   #9
Jul 2014

3·149 Posts

Right. Well thanks for those posts. I thought my thread had come to it's natural end but I'm glad it hasn't.

Please don't take this to mean I'm not interested in the latest replies, but since I wasn't expecting anymore I was busy looking into some more things.

I hope therefore it can be considered not to be without due interest in matters raised in this thread that I ask the following ;

As I understand it, A Euclidean Domain had a Euclidean Norm and a Euclidean Algorithm for division.
I'm fine with that.
What I'm confused about is that in the same way that Every Euclidean Domain is a UFD, every Field is a Euclidean Domain.

It seems logical to me that every field therefore has a Euclidean Norm and a Euclidean Algorithm so I'm totally puzzled about the fact that in Thomas Hardy's book The Theory of Numbers, a distinction is made between Euclidean Fields and Non-Euclidean fields.

For example, k(sqrt(23)), the real quadratic field is said not to be Euclidean whereas k(sqrt(2)) the real quadratic field associated with root 2, is said to be Euclidean.

Help, please.
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