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I am preparing for an exam in Algebraic number theory and I wanted to solve the following exercise:

a) Find the class group $\mathrm{Cl}(\mathcal{O}_K)$ of the field $K=\mathbb{Q}(\sqrt{-37})$.

b) Determine the complete set of the integer solutions to the equation $$x^2+37=y^3$$

I have successfully proven that $\mathrm{Cl}(\mathcal{O}_K)\cong \mathbb{Z}/2\mathbb{Z}$. Now, for the second part of the exercise, after a little investigation I have concluded that if $(x,y)$ is an integer solution then $x$ must be even and $y$ must be odd by assuming the opposite and then going modulo $8$ to arrive at a contradiction but after that I am getting lost. I know a method of examining whether there are or not integer solutions to the equation $x^2 +d=y^3$ when $\mathbb{Z}[\sqrt{d}]$ is a UFD. Although, this is not the case for our $d=-37$ since $\mathbb{Z}[\sqrt{-37}]$ is not a UFD.

Also, I have the same problem with the exercise:

-Determine the complete set of the integer solutions to the equation $$x^2+11=y^3$$

Finally, I know that the equations above are strongly related to elliptic curves but I cannot use such arguments.

richarddedekind
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Cornelius
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    See https://math.stackexchange.com/questions/3466879/x2k-y3-no-integral-solution/3466923#3466923 – GreginGre Dec 16 '19 at 18:18
  • Thanks, this was indeed very helpful. Should we mark this quastion somehow or should I delete it? Since the other question solves the problem this one seems unnesseary. – Cornelius Dec 16 '19 at 21:29
  • How do you conclude from $z^3=\pm(x-\sqrt{-k})$ ? – reuns Dec 17 '19 at 03:16
  • In my case, i.e. $k=37$, if you set $z=a+b\sqrt{-37}$ then you conclude from the fact that $\mathcal{O}_K$ has a $\mathbb{Z}$-basis ${1, \sqrt{-37}}$. – Cornelius Dec 17 '19 at 13:40
  • I had to squint, and now I have to ask to make sure: the square of $x$ plus $37$ is the cube of $y$? Did I read that correctly? – Mr. Brooks Dec 20 '19 at 00:07
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    @Mr.Brooks Yes, you read that correctly. – Cornelius Dec 20 '19 at 11:46

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