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Izv. RAN. Ser. Mat., 2001, Volume 65, Issue 3, Pages 85–122 (Mi izv337)  

Fermat's equation over the tower of cyclotomic fields

V. A. Kolyvagin


Abstract: Let $l>3$ be a prime, let $L_n=\mathbb Q(\root{l^{n+1}}\of 1 )$ let $R_n$ be the maximal real subfield of $L_n$, and let $H_n$ be the maximal $l$-subextension of $R_n$. We define effectively calculable integer-valued functions $\varphi_1(l)$, $\varphi_2(l)$ and $\varphi_3(l)$ such that $-1\leqslant \varphi_1(l)\leqslant \varphi_2(l)\leqslant \varphi_3(l)\leqslant (l-3)/2-I(l)$, where $I(l)$ is the index of irregularity of $l$. For $\varphi_1(l)\geqslant 0$ we prove the first case of Fermat's theorem for $L_{\varphi_1(l)}$, $R_{\varphi_2(l)}$$H_{\varphi_3(l)}$ and $l$. We obtain explicit lower estimates for $\varphi_1(l)$, $\varphi_2(l)$ and $\varphi_3(l)$. For regular $l$ (when $\varphi_1(l)\geqslant 1$) we prove the second case of Fermat's theorem for $L_{(l-3)/2}$ and $l$ and Fermat's theorem for $L_{\varphi_1(l)}$$R_{\varphi_2(l)}$ and $l$, generalizing the classical result on the validity of Fermat's theorem for $L_0$ and regular $l$. We also obtain some other results on solutions of Fermat's equation $x^l+y^l+z^l=0$ over $L_n$, $R_n$ and $H_n$.

DOI: https://doi.org/10.4213/im337

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English version:
Izvestiya: Mathematics, 2001, 65:3, 503–541

Bibliographic databases:

MSC: 11D41, 11R18, 11R29
Received: 07.08.2000

Citation: V. A. Kolyvagin, “Fermat's equation over the tower of cyclotomic fields”, Izv. RAN. Ser. Mat., 65:3 (2001), 85–122; Izv. Math., 65:3 (2001), 503–541

Citation in format AMSBIB
\Bibitem{Kol01}
\by V.~A.~Kolyvagin
\paper Fermat's equation over the tower of cyclotomic fields
\jour Izv. RAN. Ser. Mat.
\yr 2001
\vol 65
\issue 3
\pages 85--122
\mathnet{http://mi.mathnet.ru/izv337}
\crossref{https://doi.org/10.4213/im337}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1853367}
\zmath{https://zbmath.org/?q=an:1016.11012}
\transl
\jour Izv. Math.
\yr 2001
\vol 65
\issue 3
\pages 503--541
\crossref{https://doi.org/10.1070/IM2001v065n03ABEH000337}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33746856954}


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