Wikipedia

Chebotarev theorem on roots of unity

Not to be confused with Chebotarev's density theorem.

The Chebotarev theorem on roots of unity was originally a conjecture made by Ostrowski in the context of lacunary series.

Chebotarev was the first to prove it, in the 1930s. This proof involves tools from Galois theory and pleased Ostrowski, who made comments arguing that it "does meet the requirements of mathematical esthetics".[1] Several proofs have been proposed since,[2] and it has even been discovered independently by Dieudonné.[3]

Statement

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Let   be a matrix with entries  , where  . If   is prime then any minor of   is non-zero.

Equivalently, all submatrices of a DFT matrix of prime length are invertible.

Applications

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In signal processing,[4] the theorem was used by T. Tao to extend the uncertainty principle.[5]

Notes

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  1. ^ Stevenhagen et al., 1996
  2. ^ P.E. Frenkel, 2003
  3. ^ J. Dieudonné, 1970
  4. ^ Candès, Romberg, Tao, 2006
  5. ^ T. Tao, 2003

References

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  • Stevenhagen, Peter; Lenstra, Hendrik W (1996). "Chebotarev and his density theorem". The Mathematical Intelligencer. 18 (2): 26–37. CiteSeerX 10.1.1.116.9409. doi:10.1007/BF03027290. S2CID 14089091.
  • Frenkel, PE (2003). "Simple proof of Chebotarev's theorem on roots of unity". arXiv:math/0312398.
  • Terence Tao (2005), "An uncertainty principle for cyclic groups of prime order", Mathematical Research Letters, 12 (1): 121–127, arXiv:math/0308286, doi:10.4310/MRL.2005.v12.n1.a11, S2CID 8548232
  • Dieudonné, Jean (1970). "Une propriété des racines de l'unité". Collection of Articles Dedicated to Alberto González Domınguez on His Sixty-fifth Birthday.
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