Units of a ring

• March 2nd 2010, 05:30 PM
Brattmat
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.

• March 2nd 2010, 07:01 PM
tonio
Quote:

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