Fermat's Last Theorem in the cyclotomic integers.

2011-02-20T21:10:20Z

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.

Answer by Tauno Metsänkylä for Fermat's Last Theorem in the cyclotomic integers.

2011-02-23T19:37:25Z

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 $(x,y,z)$.

Fermat's Last Theorem in the cyclotomic integers. - MathOverflow

Fermat's Last Theorem in the cyclotomic integers.

Answer by Tauno Metsänkylä for Fermat's Last Theorem in the cyclotomic integers.