Does Fermat's Last Theorem hold for cyclotomic integers in $\mathbb{Q(\zeta_{37})}$?
The first irregular prime is 37. Does FLT(37)
$$x^{37} + y^{37} = z^{37}$$
have any solutions in the ring of integers of $\mathbb Q(\zeta_{37})$, where $\zeta_{37}$ is a primitive 37th root of unity?
Maybe it's not true, but how could I go about finding a counter-example? (for any cyclotomic ring, not necessarily 37)
This question was answered in mathoverflow. I am writing this to close up this question and making this answer as a community wiki according to MSE's guidelines. The answer is due to Tauno Metsänkylä.