# Does Fermat’s Last Theorem hold for cyclotomic integers in Q(ζ37)\mathbb{Q(\zeta_{37})}?

The first irregular prime is 37. Does FLT(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.