View source for Chebotarev theorem on roots of unity

{{short description|All submatrices of a discrete Fourier transform matrix of prime length are invertible}}
{{Distinguish|Chebotarev's density theorem}}

The '''Chebotarev theorem on roots of unity''' was originally a conjecture made by [[Alexander Ostrowski|Ostrowski]] in the context of [[Lacunary function|lacunary series]].

[[Nikolai Chebotaryov|Chebotarev]] was the first to prove it, in the 1930s. This proof involves tools from [[Galois extension|Galois theory]] and pleased [[Alexander Ostrowski|Ostrowski]], who made comments arguing that it "''does meet the requirements of mathematical esthetics''".<ref>Stevenhagen et al., 1996</ref>
Several proofs have been proposed since,<ref>P.E. Frenkel, 2003</ref> and it has even been discovered independently by [[Jean Dieudonné|Dieudonné]].<ref>J. Dieudonné, 1970</ref>

== Statement ==

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

Equivalently, all [[Submatrix|submatrices]] of a [[DFT matrix]] of prime length are invertible.

== Applications ==
In [[signal processing]],<ref>Candès, Romberg, Tao, 2006</ref> the theorem was used by [[Terence Tao|T. Tao]] to extend the [[Uncertainty principle#Signal processing|uncertainty principle]].<ref>T. Tao, 2003</ref>

== Notes ==
{{reflist}}

== References ==

*{{cite journal
 | title=Chebotarev and his density theorem
 |author1=Stevenhagen, Peter  |author2=Lenstra, Hendrik W
 | journal=The Mathematical Intelligencer
 | volume=18 |issue=2 |pages=26–37
 | year=1996
 | doi =10.1007/BF03027290
| citeseerx=10.1.1.116.9409
 |s2cid=14089091 }}
*{{cite arXiv
 | title=Simple proof of Chebotarev's theorem on roots of unity
 | author=Frenkel, PE
 | year=2003
 | eprint=math/0312398
}}

*{{citation|
journal=Mathematical Research Letters
|volume=12
|year=2005
|issue=1
|title=An uncertainty principle for cyclic groups of prime order
|pages=121–127
|author=[[Terence Tao]]
|doi=10.4310/MRL.2005.v12.n1.a11
|arxiv=math/0308286
|s2cid=8548232
}}
*{{cite journal
 | title= Une propriété des racines de l'unité
 | author=Dieudonné, Jean
 | journal=Collection of Articles Dedicated to Alberto González Domınguez on His Sixty-fifth Birthday
 | year=1970
}}

*{{cite journal
 | title=Stable signal recovery from incomplete and inaccurate measurements
 |author1=Candes, Emmanuel J |author2=Romberg Justin K |author3=Tao, Terence | journal=Communications on Pure and Applied Mathematics
 | volume=59 |issue=8
 | pages=1207–1223
 | year=2006
 | arxiv=math/0503066
 | bibcode=2005math......3066C
 | doi=10.1002/cpa.20124
|s2cid=119159284 }}
<!-- This will add a notice to the bottom of the page and won't blank it! The new template which says that your draft is waiting for a review will appear at the bottom; simply ignore the old (grey) drafted templates and the old (red) decline templates. A bot will update your article submission. Until then, please don't change anything in this text box and press "Save page". -->

[[Category:Theorems in linear algebra]]
[[Category:Theorems in algebraic number theory]]