en>Mrt3366: minor edits using AWB (8964)

2013-03-24T06:40:16Z

minor edits using AWB (8964)

New page

{{Distinguish|Chebotarev's density theorem}}

The theorem state that all [[Submatrix|submatrices]] of a [[DFT matrix]] of prime length are invertible.

The '''Chebotaryov theorem on roots of unity''' was originally a conjecture made by [[Alexander Ostrowski|Ostrowski]] in the context of [[Lacunary function|lacunary series]]. [[Nikolai Chebotaryov|Chebotaryov]] was the first to prove it, in the 1930s. This proof involves tools from [[Galois extension|Galois theory]] and did not please [[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^{2i\pi / n},n\in \mathbb{N}</math>.
If <math>n</math> is prime then any minor of <math> \Omega </math> is non-zero.

== Applications ==
For signal processing purposes,<ref>Candès, Romberg, Tao, 2006</ref> as a consequence of the '''Chebotaryov theorem on roots of unity''', [[Terence Tao|T. Tao]] stated an extension of the [[Uncertainty_principle#Signal_processing|uncertainty principle]].<ref>T. Tao, 2003</ref>

== Notes ==

<references/>

== References ==

*{{cite journal |
title=Chebotarev and his density theorem|
author=Stevenhagen, Peter and Lenstra, Hendrik W|
journal=The Mathematical Intelligencer|
volume=18|
pages=26–37|
year=1996|
publisher=Springer|
issue=2 | doi =10.1007/BF03027290
}}
*{{cite journal |
title=Simple proof of Chebotarev's theorem on roots of unity |
author=Frenkel, PE |
journal=ArXiv preprint math/0312398 |
year=2003|arxiv=math/0312398 |
bibcode=2003math.....12398F |
pages=12398
}}

*{{cite journal |
title=An uncertainty principle for cyclic groups of prime order|
author=Tao, Terence |
journal=ArXiv preprint math/0308286 |
year=2003|arxiv=math/0308286 |
bibcode=2003math......8286T |
pages=8286
}}
*{{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 |
author=Candes, Emmanuel J and Romberg, Justin K and Tao, Terence |
journal=Communications on pure and applied mathematics |
volume=59 |
pages=1207–1223 |
year=2006 |
publisher=Wiley Online Library
|arxiv=math/0503066 |
issue=8 | bibcode=2005math......3066C | last2=Romberg | last3=Tao | doi=10.1002/cpa.20124
}}


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

Maltokinase - Revision history

en>Mrt3366: minor edits using AWB (8964)