Chebotarev theorem on roots of unity

{{short description|All submatrices of a discrete Fourier transform matrix of prime length are invertible}}

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".Stevenhagen et al., 1996

Several proofs have been proposed since,P.E. Frenkel, 2003 and it has even been discovered independently by Dieudonné.J. Dieudonné, 1970

Statement

Let $\Omega$ be a matrix with entries $a_{ij} =\omega^{ij},1\leq i,j\leq n$ , where $\omega =e^{2\mathrm i\pi / n},n\in \mathbb{N}$ .

If $n$ is prime then any minor of $\Omega$ is non-zero.

Equivalently, all submatrices of a DFT matrix of prime length are invertible.

Applications

In signal processing,Candès, Romberg, Tao, 2006 the theorem was used by T. Tao to extend the uncertainty principle.T. Tao, 2003

Notes

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 }}

Category:Theorems in linear algebra

Category:Theorems in algebraic number theory