Fermat's equation over the tower of cyclotomic fields

Let $l>3$ be a prime, let $L_n=\mathbb Q\bigl(\root{l^{n+1}}\of 1\,\bigr)$ 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 $H_n$.