Show that there exist exactly two functions $f : \Bbb Q → \Bbb Q$ with the property $f(x + y) = f(x) + f(y)$ and $f(x · y) = f(x) · f(y)$ for all $x, y \in \Bbb Q$.
I am unsure how to prove that there are no more than two functions that meet the requirements. I can come up with an example $f(x) = x$ which obviously satisfy the constraints but I can't see a way to prove that there are only two functions.
Your properties are very powerful. In particular, they are so powerful that by setting the value of the function at a single point, the value of the function is determined for every rational number. To prove this, first prove it for positive naturals, then integers, then rationals.
Once you've proven this, we can inspect possible "generating" values and see what happens. For example, assume $f(1)=a$. This tells us that $f(x)=a\cdot f(x)$. There's only two ways that this equality can hold, can you spot what they are?
