I'm reading an introductory book on algebraic numbers. The author has defined the notions of a norm on $\mathbb{Z}[\zeta_n]$ by the usual formula (I give it for $n=5$):
$\alpha=f(\zeta_5)=a+b\zeta_5+c\zeta_5^2+d\zeta_5^3$ $$ N(\alpha)=f(\zeta_5)f(\zeta_5^2)f(\zeta_5^3)f(\zeta_5^4). $$
No heavy machinary of algebraic number theory has been developed at this stage of the book. An exercise in the book asks to show that $\mathbb{Z}[\zeta_5]$ is euclidean. I know that it is enough to show that $$ -\frac{1}{2}\le a,b,c,d\le\frac12\,\, \Longrightarrow\,\, N(\alpha)<1. $$
I was able to calculate the norm $$ N(\alpha)=\left(p^2+\frac{\sqrt{5} \phi}{4} q^2\right)\left(r^2+\frac{\sqrt{5} }{4 \phi }s^2\right) $$ where $$ p=a-\frac{b \phi }{2}+\frac{c+d}{2 \phi } $$ $$ q=\frac{b}{\phi }-c+d $$ $$ r=a+\frac{b}{2 \phi }-\frac{1}{2} \phi (c+d) $$ $$ s=-b \phi -c+d $$ where $\phi=\frac{1+\sqrt{5}}{2}$ is the Golden Ratio.
But how to proceed further? Simple bounding of the terms like in the case of $n=3$ does not work. We could find maximum of $N(\alpha)$ in the defined range, but not clear to me how to solve. The problem seems to be easy, because the book is for undergrads that are just starting out in algebraic numbers. I have spent a day already thinking about this question. The same exercise also asks about units of $\mathbb{Z}[\zeta_5]$. I know that $1+\zeta_5$ is a unit. Is this supposed to help to solve the problem somehow? What am I missing?