- #1
gonzo
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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?
\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?
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