- #1

gonzo

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Let

[tex]\displaystyle{\zeta = e^{{2\pi i} \over 5}}[/tex]

I need to show that [itex]Z[\zeta][/itex] is a Euclidean ring.

The only useful technique I know about is showing that given an element [itex]\epsilon \in Q(\zeta)[/itex] we can always find [itex]\beta \in Z[\zeta][/itex] such that [itex]N(\epsilon - \beta) < 1[/itex] (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?

[tex]\displaystyle{\zeta = e^{{2\pi i} \over 5}}[/tex]

I need to show that [itex]Z[\zeta][/itex] is a Euclidean ring.

The only useful technique I know about is showing that given an element [itex]\epsilon \in Q(\zeta)[/itex] we can always find [itex]\beta \in Z[\zeta][/itex] such that [itex]N(\epsilon - \beta) < 1[/itex] (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?

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