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Old 2019-07-04, 13:18   #10
Dr Sardonicus
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Feb 2017

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Originally Posted by wildrabbitt View Post
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.
In a field, every division (by any nonzero element) "comes out even" with a remainder of zero.

You're unclear on the definitions. As the term is used in Hardy and Wright, "Euclidean field" is a number field whose ring of algebraic integers has a Euclidean (division with quotient and remainder) algorithm. The remainder is either 0 or is "smaller" than the divisor. The usual function used to measure the "size" of integers is the absolute value of the norm. You might try reading The Euclidean Algorithm in Quadratic Number Fields.
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