Chebotarev theorem on roots of unity

<h2 id="statement">Statement</h2>
<p>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.
</p><p>Equivalently, all <a href="/facts/Submatrix/qa8FI4ko">submatrices</a> of a <a href="/facts/DFT_matrix/oQM4frPU">DFT matrix</a> of prime length are invertible.
</p>
<h2 id="applications">Applications</h2>
<p>In <a href="/facts/Signal_processing/Npaja6zb">signal processing</a>,<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> the theorem was used by <a href="/facts/Terence_Tao/Do68JKn6">T. Tao</a> to extend the <a href="/facts/Uncertainty_principle/UUSoSp4T">uncertainty principle</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>
</p>
<h2 id="notes">Notes</h2>

<ul><li>Stevenhagen, Peter; Lenstra, Hendrik W (1996). "Chebotarev and his density theorem". <i>The Mathematical Intelligencer</i>. 18 (2): 26–37. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.116.9409">10.1.1.116.9409</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF03027290">10.1007/BF03027290</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:14089091">14089091</a>.</li>
<li>Frenkel, PE (2003). "Simple proof of Chebotarev's theorem on roots of unity". <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/0312398">math/0312398</a>.</li></ul>
<ul><li><a href="/facts/Terence_Tao/Do68JKn6">Terence Tao</a> (2005), "An uncertainty principle for cyclic groups of prime order", <i>Mathematical Research Letters</i>, 12 (1): 121–127, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/0308286">math/0308286</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.4310%2FMRL.2005.v12.n1.a11">10.4310/MRL.2005.v12.n1.a11</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:8548232">8548232</a></li>
<li>Dieudonné, Jean (1970). "Une propriété des racines de l'unité". <i>Collection of Articles Dedicated to Alberto González Domınguez on His Sixty-fifth Birthday</i>.</li></ul>
<ul><li>Candes, Emmanuel J; Romberg Justin K; Tao, Terence (2006). "Stable signal recovery from incomplete and inaccurate measurements". <i>Communications on Pure and Applied Mathematics</i>. 59 (8): 1207–1223. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/0503066">math/0503066</a>. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2005math......3066C">2005math......3066C</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1002%2Fcpa.20124">10.1002/cpa.20124</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:119159284">119159284</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Stevenhagen et al., 1996 <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>P.E. Frenkel, 2003 <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>J. Dieudonné, 1970 <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Candès, Romberg, Tao, 2006 <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>T. Tao, 2003 <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
</ol>

Chebotarev theorem on roots of unity open-in-new

Chebotarev theorem on roots of unity