Units of a ring

**Brattmat** · March 2nd 2010, 05:30 PM

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.

Thanks in advance for any help you can give me.

**tonio** · March 2nd 2010, 07:01 PM

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.

Thanks in advance for any help you can give me.

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.

Tonio

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