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.)

  • 2
    $\begingroup$ 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. $\endgroup$ – Henning Makholm Oct 16 '16 at 15:39
  • $\begingroup$ @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. $\endgroup$ – user378947 Oct 16 '16 at 15:52
  • $\begingroup$ 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. $\endgroup$ – Asaf Karagila Oct 18 '16 at 11:44
  • $\begingroup$ $\mathbb{Q}(\sqrt{-163})$ is a PID because it is a field. :-) $\endgroup$ – lhf Oct 18 '16 at 12:38
  • $\begingroup$ @lhf Nice observation :) $\endgroup$ – 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.

  • $\begingroup$ Sorry, I couldn't respond any sooner. First of all thank you for your answer, I greatly appreciate it. My knowledge of cuadratic forms is a little basic so I apologize If I am not getting right what you mean. Are you implying that the question lacks some consistence or historical evidence? $\endgroup$ – user378947 Oct 18 '16 at 18:21
  • $\begingroup$ The question was not how Gauss did proceed - he did not know about PIDs in the first place, I'm saying that the insistence on historical correctness adds technical problems. If you use forms with odd middle coefficients then Gauss's original proof carries over easily. $\endgroup$ – franz lemmermeyer Oct 18 '16 at 18:29
  • $\begingroup$ 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)... $\endgroup$ – user378947 Oct 18 '16 at 19:42
  • $\begingroup$ ..., and one might especulate about how he would try to prove. Maybe he could prove it with the tools at his disposal, or maybe he could come up with a variant of an argument he knew, or maybe the right especulation would be that he lacked too much techniques and results that only today we know and because of that it can be argued that there actually no evidence to say anything about it... Frankly y don't know, and that's why I am asking. What I was trying to say is that, maybe because I do not know too mucho about quadratic forms, but I didn't get any of that clear from your answer. ... $\endgroup$ – user378947 Oct 18 '16 at 19:55
  • $\begingroup$ I don't see how much trouble could be to Gauss to think about quadratic forms other than those with even middle term (again this might be because of my ignorance), and I don't know if you think that it is feasible that he could give some or another argument, ... Sorry for the extension. $\endgroup$ – user378947 Oct 18 '16 at 20:03

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