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A note on the cyclotomic polynomial

  • I. J. Schoenberg (a1)


The n-th roots of unity 1, ω, …, ωn-1, where ω = exp (2πi\n), are linearly dependent in the field Q of rationals since, for instance, their sum vanishes. We are here concerned with the linear relations between them with integral coefficients. Let U denote the vector space of elements u = (u0, …, un−1) over Q and let N be the subspace of elements u defined by the relation u0+u1ω+…+un−1ωn−1=0. (1)



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1. Smith, H. J. S., “On systems of linear indeterminate equations and congruences”, No. XII in Collected mathematical papers, vol. 1, (Oxford, 1894).
2. Tschebotaröw, N. and Schwerdtfeger, H., Grundzüge der Galoisschen Theorie, (Groningen, 1950).
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