An integral domain $R$ is said to be Euclidean if it admits some Euclidean norm: i.e., a function $N: R \rightarrow \mathbb{N} = \mathbb{Z}^{\geq 0}$ such that: for all $x, y \in R …

The question is in the title, but employs some private terminology, so I had better explain. Let $R$ be an integral domain with fraction field $K$, and write $R^{\bullet}$ for $R …

Hi all, Z[cis(2pi/3)]:={a+bw|a,b are integers,w=cos(2pi/3)+isin(2pi/3)}. I want to show that Z[w] is euclidean , I tried doing it in similar fashion to the way we show (at least t …

Can anyone suggest a good place to read up on the number theoretic properties of and techniques for $\mathbb{Z}[1/p]$, (that is, rational numbers with only powers of a prime $p$ in …