# “How would Gauss proceed?”

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.)

• Presumably, in the class that exam was for, how Gauß actually did proceed with a closely similar problem had been discussed and dissected at length. – Henning Makholm Oct 16 '16 at 15:39
• @HenningMakholm It seems to me like a difficult question, so I supposed that too. However, I am just curious on the evidence to tell how would Gauss proceed, and on the argument too. – user378947 Oct 16 '16 at 15:52
• When I was a freshman, in the discrete math exam we were asked to find a sequence of natural numbers $a_n$, such that $\lim a_{n+1}/a_n=\sqrt3$. Of course, this is a very difficult problem if you have never tackled it before. But we were taught the proof that for the Fibonacci sequence, the limit is $\varphi$, so all you had to do was to reverse engineer the proof. The point is, that often a question might seem hard or nearly impossible, because you weren't present in the class were they taught exactly the necessary tools to deal with the question. – Asaf Karagila Oct 18 '16 at 11:44
• $\mathbb{Q}(\sqrt{-163})$ is a PID because it is a field. :-) – lhf Oct 18 '16 at 12:38
• @lhf Nice observation :) – user378947 Oct 18 '16 at 18:06

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.
• Yes, the question is how Gauss might proceed; I am not asking for historical correctness, I am asking about historical feasibleness. Based on his works one might get a more or less accurate idea of his knowledge of mathematics. From here one can imagine that he finds the problem of determining if $\mathbb{Q}(\sqrt{-163})$ is a PID (or that every closed-under-addition set that is absorbent with respect to the product can be generated by one single element)... – user378947 Oct 18 '16 at 19:42