The classical criteria of Kummer, Mirimanov and Vandiver for the validity of the first case of Fermat's theorem for the field Q of rationals and prime exponent l are generalized to the field Q(1~(1/l)) 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 Q(1~(1/l)) and exponent l on condition that l~2 does not divide 2~l - 2.
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