Hints for proving some algebraic integer rings are Euclidean In my book - Algebraic Number Theory and Fermat's Last Theorem Ian Stewart, David Tall - there is the exercise:


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*Prove that the ring of integers of $\mathbb{Q}(\zeta_5)$ (the 5'th cyclotomic ring) is Euclidean.


I can prove that the integers of $\mathbb{Q}(\sqrt{-1})$ are Euclidean by a geometric argument but this doesn't work for the first problem since pentagons aren't in a lattice.
If anyone could give me a hint, thank you.

HINT $\ \ $ It is norm Euclidean, i.e. the absolute value of the norm serves as a Euclidean function. For a nice survey see Lemmermeyer: The Euclidean algorithm in algebraic number fields.
