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 In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory t ...

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

\Omega

be a matrix with entries

a_ =\omega^,1\leq i,j\leq n

, where

\omega =e^,n\in \mathbb

. If

n

is prime then any minor of

\Omega

is non-zero. Equivalently, all

submatrices In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, \begin ...

of a

DFT matrix In applied mathematics, a DFT matrix is an expression of a discrete Fourier transform (DFT) as a transformation matrix, which can be applied to a signal through matrix multiplication. Definition An ''N''-point DFT is expressed as the multiplicati ...

of prime length are invertible.

Applications

signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...

, the theorem was used by T. Tao to extend the

uncertainty principle In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...

.T. Tao, 2003

Notes

References

* * * * *{{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 Theorems in linear algebra Theorems in algebraic number theory