ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

3. Number Theory and Arithmetic Algebraic Geometry

1991 MS Classification: 11 Canales, Mónica del Pilar, Universidad Austral de Chile, Chile Higher levels of Z/p^{l}Z and cyclotomy In this work are first studied the d^{th} levels of the prime finite fields F_{p}, defined as

s_{d}(F_{p}) = min {s | −1 = a_{1}^{d} + ⋯ + a_{s}^{d}, a_{i} ∈ F_{p}}

where p ≠ 2 is a prime number, d ≥ 2 is an even rational integer and d | p−1 without loss of generality. The main result establishes that s_{d}(F_{p}) is the minimal s for which (1+s(p−1)/d)α_{s}−(s+1)α_{s+1} ≠ 0, where α_{i}, 1 ≤ i ≤ d, are the coefficients of the Gauss period equation of degree d, resolvent of the cyclotomic equation X^{p} − 1 = 0. Similarly, s_{d}(Z/ p^{l}Z) for l ≥ 1 is obtained in terms of the coefficients of generalized period equations, while s_{d}(Z/2^{l}Z), l ≥ 1 is found in an elementary way. References: E. Becker, M. del P. Canales, Higher levels of Z/p^{l}Z and period equations (submitted)