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". Several proofs have been proposed since, and it has even been discovered independently by Dieudonné.
Statement
Let Ω {\displaystyle \Omega } be a matrix with entries a i j = ω i j , 1 ≤ i , j ≤ n {\displaystyle a_{ij}=\omega ^{ij},1\leq i,j\leq n} , where ω = e 2 i π / n , n ∈ N {\displaystyle \omega =e^{2\mathrm {i} \pi /n},n\in \mathbb {N} } . If n {\displaystyle n} is prime then any minor of Ω {\displaystyle \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
Notes
- 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.
- 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. S2CID 119159284.