How do mathematicians know that what they are researching has not been already known for $200$ years? Obviously, if they are researching something that is cutting edge it is not a problem, but if one is investigating a problem in a very old field like Euclidean Geometry then this could be a problem.
I am interested in the problem, how many prime polynomials of degree $n$ are there in $\mathbb Z/p\mathbb Z[x]$? When I google this problem, I get no relevant results. However, for all I know, Gauss solved this problem. But how do I find out?