Geodesic
On the first case of Fermat's theorem for cyclotomic fields
Izvestiya. Mathematics , Tome 63 (1999) no. 5, pp. 983-994.

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The classical criteria of Kummer, Mirimanov and Vandiver for the validity of the first case of Fermat's theorem for the field $\mathbb Q$ of rationals and prime exponent $l$ are generalized to the field $\mathbb Q(\root l\of 1)$ and exponent $l$. As a consequence, some simpler criteria are established. For example, the validity of the first case of Fermat's theorem is proved for the field $\mathbb Q(\root l\of 1)$ and exponent $l$ on condition that $l^2$ does not divide $2^l-2$. 
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V. A. Kolyvagin. On the first case of Fermat's theorem for cyclotomic fields. Izvestiya. Mathematics , Tome 63 (1999) no. 5, pp. 983-994. http://geodesic.mathdoc.fr/item/IM2_1999_63_5_a4/

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