1. Units of a ring

Hi all, I'm stuck on this question unfortunately and could use some help:

Show the group of units (Z[√6])× in the ring Z[√6] is infinite by exhibiting infinitely many units in each of the rings.

I've had a good look through my notes as well as books and online but I'm still unsure how to go about solving this.

2. Originally Posted by Brattmat
Hi all, I'm stuck on this question unfortunately and could use some help:

Show the group of units (Z[√6])× in the ring Z[√6] is infinite by exhibiting infinitely many units in each of the rings.

I've had a good look through my notes as well as books and online but I'm still unsure how to go about solving this.

An element $\displaystyle a+b\sqrt{6}\in\mathbb{Z}[\sqrt{6}]$ is a unit iff its norm is $\displaystyle \pm1\Longleftrightarrow N(a+b\sqrt{6})=a^2-6b^2=\pm 1$ , so for example $\displaystyle 5\pm 2\sqrt{6}$ are units...but
also $\displaystyle \left(5\pm 2\sqrt{6}\right)^n\,,\,\,n\in\mathbb{N}$ are units. The only thing left to do is to show that $\displaystyle \left\{\left(5+2\sqrt{6}\right)^n\right\}_{n=1}^\i nfty$ , for example, is an infinite set.