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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The orders of solutions of the Kummer system of congruences


Author: Ladislav Skula
Journal: Trans. Amer. Math. Soc. 343 (1994), 587-607
MSC: Primary 11D41; Secondary 11B68
MathSciNet review: 1196218
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Abstract: A new method concerning solutions of the Kummer system of congruences (K) (modulo an odd prime l) is developed. This method is based on the notion of the Stickelberger ideal. By means of this method a new proof of Pollaczek's and Morishima's assertion on solutions of (K) of orders 3, 6 and 4 $ \bmod\; l$ is given. It is also shown that in case there is a solution of $ (K)\, \not\equiv \, 0, \pm 1\;\pmod l$, then for the index of irregularity $ i(l)$ of the prime l we have $ i(l) \geq [\sqrt[3]{{l/2}}]$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1994-1196218-5
PII: S 0002-9947(1994)1196218-5
Keywords: Kummer system of congruences, the first case of Fermat's Last Theorem, Stickelberger ideal, index of irregularity of a prime
Article copyright: © Copyright 1994 American Mathematical Society