I am looking for universities with a graduate program in the United States. I started with Princeton (dreaming is free:) and learned that in order to begin working on their theses the students have to spend a year studying there, and pass a General Examination. I got curious about how difficult this tests would be. They have the tradition to post records online of past exams so, after nosing for a while, I found this question by Professor Sarnak in a 2008 exam; he just had asked the student how would he prove that $\mathbb{Q}(\sqrt{-163})$ is a principal ideal domain when he asks him "How would Gauss proceed?".
At first it sounded to me like an unfair and ridiculous question, but who am I to contradict Professor Sarnak. Do you know if there are actually enough reason to tell how would Gauss proceed? And if that is the case, what would be his argument?
(I know that "enough" is not very precise and that it can be thought as a matter of opinion. I will content myself with plausible reasons motivating a specific argument Gauss could have used to answer the above question.)
Gauß proved that ${\mathbb Z}[i]$ is a PID using the fact that the class number of forms with discriminant $-4$ is $1$. On the other hand, Gauß only considered quadratic forms with even middle coefficient, so in the case of discriminant $-163$ he would have been forced to use the fact that the number of classes of forms with discriminant $-163$ is $3$, and the rest of the proof would then require additional arguments. I don't think, however, that this was the point of the question, which was aimed at getting binary quadratic forms as an answer.