Chebotarev theorem on roots of unity explained

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

Let

\Omega be a matrix with entries a _{ij} = \omega ^{ij} , 1 \leq i,j\leq n , where \omega = e ^{2i\pi/} ,n\in N . If n is prime then any minor of \Omega is non-zero.

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

Applications

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

References

• Chebotarev and his density theorem . Stevenhagen, Peter . Lenstra, Hendrik W . The Mathematical Intelligencer . 18 . 2 . 26–37 . 1996 . 10.1007/BF03027290. 10.1.1.116.9409 . 14089091 .
• Simple proof of Chebotarev's theorem on roots of unity . Frenkel, PE . 2003 . math/0312398.
  • Une propriété des racines de l'unité . Dieudonné, Jean . Collection of Articles Dedicated to Alberto González Domınguez on His Sixty-fifth Birthday . 1970.
• Stable signal recovery from incomplete and inaccurate measurements . Candes, Emmanuel J . Romberg Justin K . Tao, Terence . Communications on Pure and Applied Mathematics . 59 . 8 . 1207–1223 . 2006 . math/0503066 . 2005math......3066C . 10.1002/cpa.20124. 119159284 .

Notes and References

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