Ring Theory: Proving $\mathbb{Z} [ \sqrt{2} ]$ has Infinitely Many Units

[QuantumJG]

Show $$\mathbb{Z} [ \sqrt{2} ]$$ = $$\{ a + b \sqrt{2} | a,b \in \mathbb{Z} \}$$ has infinitely many units.

I started by taking an element:

$$a + b \sqrt{2} \in \mathbb{Z} [ \sqrt{2} ]$$

and finding an inverse

$$\left( a + b \sqrt{2} \right) ^{-1}$$

such that the product gives zero and tried to show any element works. But I'm not sure about doing this.

 

[deluks917]

The product should give 1.

 

[Dick]

And I found the inverse and I didn't see an infinite number of units. Z is the integers, right? What did you get for the inverse?

 

[Dick]

And I found the inverse and I didn't see an infinite number of units. Z is the integers, right? What did you get for the inverse?

Ooops. My mistake. There are more units than just 1 and -1. Can you find some? Once you've found one that isn't 1 or -1, can you think of a simple way to use it to generate more?