Chebotaryov theorem on roots of unity

The theorem state that all submatrices of a DFT matrix of prime length are invertible.

The Chebotaryov theorem on roots of unity was originally a conjecture made by Ostrowski in the context of lacunary series. Chebotaryov was the first to prove it, in the 1930s. This proof involves tools from Galois theory and did not please Ostrowski, who made comments arguing that it "does not 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 ${\displaystyle a_{ij}=\omega ^{ij},1\leq i,j\leq n}$, where ${\displaystyle \omega =e^{2i\pi /n},n\in \mathbb {N} }$. If $n$ is prime then any minor of $\Omega$ is non-zero.

Applications

For signal processing purposes,^[4] as a consequence of the Chebotaryov theorem on roots of unity, T. Tao stated an extension of the uncertainty principle.^[5]

Notes

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

References

• Stevenhagen, Peter; Lenstra, Hendrik W (1996). "Chebotarev and his density theorem". The Mathematical Intelligencer. 18 (2): 26–37. doi:10.1007/BF03027290.
• Frenkel, PE (2003). "Simple proof of Chebotarev's theorem on roots of unity". arXiv:math/0312398.

• Tao, Terence (2003). "An uncertainty principle for cyclic groups of prime order". arXiv:math/0308286.
• Dieudonné,Jean (1970). "Une propriété des racines de l'unité". Collection of articles dedicated to Alberto González Domınguez on his sixty-fifth birthday.

• Candes, Emmanuel J; Romberg Justin K; Tao, Terence (2006). "Stable signal recovery from incomplete and inaccurate measurements". Communications on Pure and Applied Mathematics. 59 (8): 1207–1223. arXiv:math/0503066. Bibcode:2005math......3066C. doi:10.1002/cpa.20124.