Euclidean Ring of Z[\zeta]: Unconventional Technique

[gonzo]

Let
$$\displaystyle{\zeta = e^{{2\pi i} \over 5}}$$
I need to show that $Z[\zeta]$ is a Euclidean ring.

The only useful technique I know about is showing that given an element $\epsilon \in Q(\zeta)$ we can always find $\beta \in Z[\zeta]$ such that $N(\epsilon - \beta) < 1$ (using the standard norm for the euclidean function).

This usually involves finding a general expression for the norm and then saying that you can choose beta such that the difference of each basis element is less than 1/2, and then showing that this means you can also get the norm less than 1.

However, the expression I got for the norm here didn't seem to lend itself to this method.

Any suggestions on how to do this?

 

[matt grime]

show it has a thingummmy - euclidean function, can't remember the precise name, that might help.

 

[gonzo]

Of course, but that's the point. The problem is I can't show the Norm is less than one if the coefficients are less than 1/2 and don't know any other techniques.