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          Institute: MPI für Mathematik     Collection: Export Zagier     Display Documents



ID: 744924.0, MPI für Mathematik / Export Zagier
On the coefficients of the minimal polynomials of Gaussian periods.
Authors:Gupta, S.; Zagier, Don
Date of Publication (YYYY-MM-DD):1993
Title of Journal:Mathematics of Computation
Volume:60
Issue / Number:201
Start Page:385
End Page:398
Audience:Not Specified
Intended Educational Use:No
Abstract / Description:Using standard notation let ℓ be a prime, m a divisor of ℓ-1, ω= ζ+ ζ\sp λ+ ⋅s+ ζ\spλ\spm-1, where ζ= e\sp2π i/ℓ and λ is a primitive m-th root of unity \text mod ℓ, so that ω generates a subfield k of \bbfQ (ζ) of degree (ℓ-1) /m. \par To follow the authors' abstract. The paper considers the reciprocal minimum polynomial F\sbℓ,m (X)= N\sbk/ \bbfQ (1-ω X) of ω over \bbfQ and shows that for fixed m and all N, F\sbℓ,m (X)\equiv (B\sb m (x)\sp ℓ/ (1-mX) )\sp1/m\bmod X\sp N for all but finitely many ``exceptional primes'' ℓ (depending on m and N), where B\sb m (X) is a power series in X defined only on m. Further a method of computing this exceptional set of primes is given. \par It is worth noting that the cases m=3,4 of some of the results presented were proved by D. and E. Lehmer and the case m=p by S. Gurak. The case m=2 was essentially known to Gauss.
External Publication Status:published
Document Type:Article
Communicated by:nn
Affiliations:MPI für Mathematik
Identifiers:LOCALID:111
ISSN:0025-5718
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