It's my first question here, my name is Massimiliano. How can I prove that if $a/b$ is a perfect square then $ab$ is also a perfect square without using the unicity of prime decomposition?
My work: If $a/b$ is a perfect square then there exists $c$ such that $c^2 = a/b$ so $c^2b^2 = (cb)^2 = ab$, and therefore $ab$ is a perfect square.
But (to the contrary) if $c^2 = ab$ and I divide by $b^2$ I obtain $(c/b)^2 = a/b$. But I don't know if $c/b$ is actually an integer.
Thank you for help!