Is this group cyclic?

Question

Is this group cyclic, and what would be its generator?

where H = {a+b(sqrt(2)) element of R | a, b element of Z}

I know that in order for a group to be cyclic if the generator equals the group, but I don't know how to apply this concept, or how to find a generator.

Thanks.

Answer 1

Suppose that there is a generator $a+b\sqrt 2$. Then it generates $1$ and $\sqrt 2$, that is, there exist $m$ and $n$ such that $$m(a+b\sqrt 2)=1$$ and $$n(a+b\sqrt 2)=\sqrt 2$$

This implies that $am=1$ (hence $a\neq 0$) and $an=0$; thus, $n=0$, a contradiction.

Answer 2

No, if this group were cyclic it follows that $$ (a+b \sqrt{2})^n=a'+b'\sqrt{2}=1 $$ for some integer $a,b,a',b',n$. But it is imposible since $\sqrt{2}$ is irrational.