One looks at a certain sum G involving the p-th roots of unity (where p is a prime number), called the quadratic gaussian sum. It is easy to see that G^2=p, which means that G itself is either the positive or the negative square root of p. Which one ? It took Gauss many years to find the answer and to prove the result. Since then some other proofs of this result have been given, and it has become the central example of what is called the root number of an L-function. So the result is very important.