Prosolvable group

<h2 id="examples">Examples</h2>
<ul><li>Let <i>p</i> be a <a href="/facts/Prime_number/L20FLkgn">prime</a>, and denote the <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> of <a href="/facts/P-adic_numbers/SDgDbkLF">p-adic numbers</a>, as usual, by 
  
    
      
        
          
            Q
          
          
            p
          
        
      
    
    {\displaystyle \mathbf {Q} _{p}}
  
. Then the <a href="/facts/Galois_group/A0p35p5c">Galois group</a> 
  
    
      
        
          Gal
        
        (
        
          
            
              
                Q
              
              ¯
            
          
          
            p
          
        
        
          /
        
        
          
            Q
          
          
            p
          
        
        )
      
    
    {\displaystyle {\text{Gal}}({\overline {\mathbf {Q} }}_{p}/\mathbf {Q} _{p})}
  
, where 
  
    
      
        
          
            
              
                Q
              
              ¯
            
          
          
            p
          
        
      
    
    {\displaystyle {\overline {\mathbf {Q} }}_{p}}
  
 denotes the <a href="/facts/Algebraic_closure/yjewG1mM">algebraic closure</a> of 
  
    
      
        
          
            Q
          
          
            p
          
        
      
    
    {\displaystyle \mathbf {Q} _{p}}
  
, is prosolvable. This follows from the fact that, for any finite <a href="/facts/Galois_extension/52fzNQbv">Galois extension</a> 
  
    
      
        L
      
    
    {\displaystyle L}
  
 of 
  
    
      
        
          
            Q
          
          
            p
          
        
      
    
    {\displaystyle \mathbf {Q} _{p}}
  
, the Galois group 
  
    
      
        
          Gal
        
        (
        L
        
          /
        
        
          
            Q
          
          
            p
          
        
        )
      
    
    {\displaystyle {\text{Gal}}(L/\mathbf {Q} _{p})}
  
 can be written as <a href="/facts/Semidirect_product/Uqf87k1I">semidirect product</a> 
  
    
      
        
          Gal
        
        (
        L
        
          /
        
        
          
            Q
          
          
            p
          
        
        )
        =
        (
        R
        ⋊
        Q
        )
        ⋊
        P
      
    
    {\displaystyle {\text{Gal}}(L/\mathbf {Q} _{p})=(R\rtimes Q)\rtimes P}
  
, with 
  
    
      
        P
      
    
    {\displaystyle P}
  
 cyclic of order 
  
    
      
        f
      
    
    {\displaystyle f}
  
 for some 
  
    
      
        f
        ∈
        
          N
        
      
    
    {\displaystyle f\in \mathbf {N} }
  
, 
  
    
      
        Q
      
    
    {\displaystyle Q}
  
 <a href="/facts/Cyclic_group/JL5zfFuL">cyclic</a> of order dividing 
  
    
      
        
          p
          
            f
          
        
        −
        1
      
    
    {\displaystyle p^{f}-1}
  
, and 
  
    
      
        R
      
    
    {\displaystyle R}
  
 of 
  
    
      
        p
      
    
    {\displaystyle p}
  
-power order. Therefore, 
  
    
      
        
          Gal
        
        (
        L
        
          /
        
        
          
            Q
          
          
            p
          
        
        )
      
    
    {\displaystyle {\text{Gal}}(L/\mathbf {Q} _{p})}
  
 is solvable.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a></li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Galois_theory/54wscJfj">Galois theory</a></li></ul>

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

<ol>
<li id="fn:1"><p>Boston, Nigel (2003), The Proof of Fermat's Last Theorem (PDF), Madison, Wisconsin, USA: University of Wisconsin Press <a href="http://psoup.math.wisc.edu/~boston/869.pdf" target="_blank">http://psoup.math.wisc.edu/~boston/869.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Prosolvable group open-in-new

Prosolvable group