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Abstract

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.