# Find all finite subsets of $\mathbb{C}$ closed to multiplication.

Find all finite subsets of $$\mathbb{C}$$ such that they are closed to multiplication.

Hint: calculate the powers of $$\sqrt2+\sqrt3$$ until they become a linear combination of two square roots.

I know that since $$|a \cdot b| = |a| \cdot |b|,$$ these subsets are all subsets of the unit circle. Also, I've concluded that these subsets consist of numbers $$e^{i \phi}$$ where $$2 \pi | \phi$$ because $$e^{i \phi_1} \cdot e^{i \phi_2} = e^{i (\phi_1+\phi_2)}$$ so these numbers always stay in a closed set.

I'm not sure how to formally prove this statement and if these are only possible solutions. Also, I'm not sure how the given hint helps me.

Any hints would be appreciated!

Hint The group generated by $$e^{iu}$$ is dense if $$u$$ is different of $${p\over q}\pi$$ where $$p,q$$ are integers. This follows from the fact a subgroup of the real line generated by a real number $$a$$ is dense if $$a$$ is not rational.
You have to add $$0$$ to the finite groups of the unit circle eventually.