Question

Is the additive group $G=\{a+b\sqrt{2}|a,b\in Z\}$cyclic?

Short answer

It is proved that it is not cyclic.

Step-by-step solution

Prove by contradiction

Let it be cyclic.

Then, there is a generator of the form$x+y\sqrt{2}$.

There would exist integers a, b such that,

$a(x+y\sqrt{2})=1\text{and}b(x+y\sqrt{2})=\sqrt{2}$.

This implies,

$$\begin{gathered} ax=1 \\ ay=0 \\ bx=0 \\ by=1 \end{gathered}$$

which is impossible to satisfy.

Conclusion

Hence, our supposition was wrong.

The group is not cyclic.