Finite extensions of local fields

<h2 id="unramified-extension">Unramified extension</h2>
<p>Let 
  
    
      
        L
        
          /
        
        K
      
    
    {\displaystyle L/K}
  
 be a <a href="/facts/Finite_extension/hY5LgHYk">finite</a> <a href="/facts/Galois_extension/52fzNQbv">Galois extension</a> of nonarchimedean local fields with finite residue fields 
  
    
      
        ℓ
        
          /
        
        k
      
    
    {\displaystyle \ell /k}
  
 and <a href="/facts/Galois_group/A0p35p5c">Galois group</a> 
  
    
      
        G
      
    
    {\displaystyle G}
  
. Then the following are equivalent.
</p>
<ul><li>(i) 
  
    
      
        L
        
          /
        
        K
      
    
    {\displaystyle L/K}
  
 is unramified.</li>
<li>(ii) 
  
    
      
        
          
            
              O
            
          
          
            L
          
        
        
          /
        
        
          
            p
          
        
        
          
            
              O
            
          
          
            L
          
        
      
    
    {\displaystyle {\mathcal {O}}_{L}/{\mathfrak {p}}{\mathcal {O}}_{L}}
  
 is a field, where 
  
    
      
        
          
            p
          
        
      
    
    {\displaystyle {\mathfrak {p}}}
  
 is the maximal ideal of 
  
    
      
        
          
            
              O
            
          
          
            K
          
        
      
    
    {\displaystyle {\mathcal {O}}_{K}}
  
.</li>
<li>(iii) 
  
    
      
        [
        L
        :
        K
        ]
        =
        [
        ℓ
        :
        k
        ]
      
    
    {\displaystyle [L:K]=[\ell :k]}
  
</li>
<li>(iv) The <a href="/facts/Inertia_subgroup/UHXcYyGh">inertia subgroup</a> of 
  
    
      
        G
      
    
    {\displaystyle G}
  
 is trivial.</li>
<li>(v) If 
  
    
      
        π
      
    
    {\displaystyle \pi }
  
 is a <a href="/facts/Uniformizing_element/DaHoTBSH">uniformizing element</a> of 
  
    
      
        K
      
    
    {\displaystyle K}
  
, then 
  
    
      
        π
      
    
    {\displaystyle \pi }
  
 is also a uniformizing element of 
  
    
      
        L
      
    
    {\displaystyle L}
  
.</li></ul>
<p>When 
  
    
      
        L
        
          /
        
        K
      
    
    {\displaystyle L/K}
  
 is unramified, by (iv) (or (iii)), <i>G</i> can be identified with 
  
    
      
        Gal
        ⁡
        (
        ℓ
        
          /
        
        k
        )
      
    
    {\displaystyle \operatorname {Gal} (\ell /k)}
  
, which is finite <a href="/facts/Cyclic_group/JL5zfFuL">cyclic</a>.
</p><p>The above implies that there is an <a href="/facts/Equivalence_of_categories/k6uzqAcy">equivalence of categories</a> between the finite unramified extensions of a local field <i>K</i> and finite <a href="/facts/Separable_extension/9KBvXmz8">separable extensions</a> of the residue field of <i>K</i>.
</p>
<h2 id="totally-ramified-extension">Totally ramified extension</h2>
<p>Again, let 
  
    
      
        L
        
          /
        
        K
      
    
    {\displaystyle L/K}
  
 be a finite Galois extension of nonarchimedean local fields with finite residue fields 
  
    
      
        l
        
          /
        
        k
      
    
    {\displaystyle l/k}
  
 and Galois group 
  
    
      
        G
      
    
    {\displaystyle G}
  
. The following are equivalent.
</p>
<ul><li>
  
    
      
        L
        
          /
        
        K
      
    
    {\displaystyle L/K}
  
 is totally ramified.</li>
<li>
  
    
      
        G
      
    
    {\displaystyle G}
  
 coincides with its inertia subgroup.</li>
<li>
  
    
      
        L
        =
        K
        [
        π
        ]
      
    
    {\displaystyle L=K[\pi ]}
  
 where 
  
    
      
        π
      
    
    {\displaystyle \pi }
  
 is a root of an <a href="/facts/Eisenstein_polynomial/ETY6bUAr">Eisenstein polynomial</a>.</li>
<li>The norm 
  
    
      
        N
        (
        L
        
          /
        
        K
        )
      
    
    {\displaystyle N(L/K)}
  
 contains a uniformizer of 
  
    
      
        K
      
    
    {\displaystyle K}
  
.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Abhyankar%27s_lemma/nHkWoDol">Abhyankar's lemma</a></li>
<li><a href="/facts/Unramified_morphism/fvUygk3n">Unramified morphism</a></li></ul>

<ul><li><a href="/facts/J._W._S._Cassels/AC33eVoA">Cassels, J.W.S.</a> (1986). <a href="https://books.google.com/books?id=UY52SQnV9w4C&q=%22finite+extension%22"><i>Local Fields</i></a>. London Mathematical Society Student Texts. Vol. 3. <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-31525-5. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0595.12006">0595.12006</a>.</li>
<li>Weiss, Edwin (1976). <a href="https://books.google.com/books?id=S38pAQAAMAAJ&q=%22finite+extension%22"><i>Algebraic Number Theory</i></a> (2nd unaltered ed.). <a href="/facts/Chelsea_Publishing/fr5gotCt">Chelsea Publishing</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8284-0293-0. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0348.12101">0348.12101</a>.</li></ul>

Finite extensions of local fields open-in-new

Finite extensions of local fields