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So I started examining FLT for $n=3$ with my professor, using properties of the Eisenstein integers. The professor made a diversion and spent a whole lecture proving that there are exactly two types of ideals in $\mathbb Z[ \sqrt{ -5}]$, but we didn't make a connection to the FLT, and that left me wondering: Where are rings of the form $\mathbb Z[\sqrt d]$ useful? Do they help us solve some other kind of Diophantine equation?

luxerhia
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JBuck
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  • These can also be used to solve some Mordell equations –  Aug 07 '20 at 09:42
  • @rain1 I see, so they aren't much related to FLT? – JBuck Aug 07 '20 at 09:45
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    I am the wrong person to describe it, but I would see they are very much related to FLT given that Kummer used "ideal numbers" (a precursor to the notion of an ideal) to prove FLT for regular primes. This was a much used avenue when proving non-existence of solutions to $x^p+y^p=z^p$ for a fixed prime $p$ (but for thousands of different $p$). – Jyrki Lahtonen Aug 07 '20 at 10:02
  • they can be applied to some specific cases of FLT absolutely –  Aug 07 '20 at 10:51
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    https://math.stackexchange.com/a/18660/581023 a very cool example –  Aug 07 '20 at 10:52

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