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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)

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    @awllower, at present I have no ideas how to approach this problem.2011-04-07

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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.