Fermat's Last Theorem in the cyclotomic integers. Kummer proved that there are no non-trivial solutions to the Fermat equation FLT(n): $x^n + y^n = z^n$ with $n > 2$ natural and $x,y,z$ elements of a regular cyclotomic ring of integers $K$.
I am looking for non-trivial solutions to the Fermat equation FLT(p) in the cyclotomic integer ring $\mathbb{Z}[\zeta_{p}]$ for irregular primes p or any information about how the solutions must be (as a step toward constructing them).
George Lowther pointed out in an earlier discussion that by Kolyvagin's criterion any solution in $\mathbb{Z}[\zeta_{37}]$ must be in the second case.
 A: This answer is a bit late; sorry for that.
Kummer's proof of the nonsolvability of $x^p + y^p = z^p$ for regular primes $p$
used “ideal numbers” (in present-day language: ideals) and was intact, at least
basically. Hilbert in his Zahlbericht gave a modified proof. Both proofs cover not
only rational integers but also numbers in $\mathbb{Z}[\zeta_p]$. On the other
hand, Kummer’s second result concerning irregular primes that satisfy certain
additional conditions covers just the rational integers (although Hilbert, in the very
last section of Zahlbericht, erroneously says that Kummer had proven this result for
$\mathbb{Z}[\zeta_p]$ as well). Thus one cannot exclude the possibility that there
is indeed a solution $(x,y,z)$ for $p=37$. And because of "Kolyvagin's criterion"
about $(2^{37}-2)/37$, this solution must belong to the second case,
that is, at least one of these three numbers $x,y,z$ in $\mathbb{Z}[\zeta_{37}]$
must have a common factor with $37$ (as mentioned by George Lowther).
By the way, this criterion was also proven by Taro Morishima in 1935 (Japan. J.
Math. 11, 241-252, Satz 1; but warning: Satz 2 or at least its proof is incorrect since
it is based on some incorrect result of Vandiver).
I don’t know how to find such a solution $(x,y,z)$.
A: Very late response but since it is still unresolved, I will answer your question. By Tauno's answer, any solution must belong to the second case, which in $\mathbb{Z}[\zeta]$ looks like $x^p+y^p=z^p$ with $1-\zeta \mid z$ and $x,y$ both coprime with $1-\zeta$. The second case also has no solutions in this ring by another criteria of Kolyvagin (from a different paper: " Fermat Equations over Cyclotomic Fields"). 
 The general criteria is a bit involved to write up here but the prime $p=37$ satisfies a simpler criteria (which applies to both the first and second case): 1) If the index of irregularity $=1$ with $p \mid B_i$ and 2) there is a prime $l \equiv 1 (\text{mod} \ p)$ for which $x^p+y^p=z^p$ has only trivial solutions modulo $l$ and $(1-\zeta(l))^{(l-3)}$, $(1-\zeta(l))^{(i)}$ are not $p$-th powers modulo $l$, then the Fermat equation has no solutions in $\mathbb{Q}(\zeta_p)$. Here $\zeta(l)$ is a primitive $p$-th root of unity modulo $l$ (which exists since $p \mid (l-1)$) and $a^{(j)}$ is the $j$-th component of $a \in \mathbb{F}_l^{\times}/(\mathbb{F}_{l}^{\times})^p$ relative to the Eigenspace decomposition of this group for the operators $\varepsilon_j = -\sum_{a=1}^{p-1}a^j\sigma_{a}^{-1}$. 
Since the index of irregularity for $37$ is $1$ with $37 \mid B_{32}$ and $x^{37}+y^{37}=z^{37}$ has only trivial solutions modulo $l=149=1+4\times 37$, then by taking $\zeta(149)=16$, we compute that $(1-16)^{(34)}$ and $(1-16)^{(32)}$ are not $37$-th powers modulo $149$. Therefore $p=37$ satisfies the criteria.
