On the first case of Fermat's theorem for cyclotomic fields

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 $2^l-2$.