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)

$$x^{37} + y^{37} = z^{37}$$

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)

Answer

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