The first irregular prime is 37. Does FLT(37)

x37+y37=z37

have any solutions in the ring of integers of Q(ζ37), where ζ37 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)

**Answer**

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.

**Attribution***Source : Link , Question Author : quanta , Answer Author :
3 revs*