Profinite integer

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a profinite integer is an element of the <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> (sometimes pronounced as zee-hat or zed-hat)

Z
              
              ^
            
          
        
        =
        
          
            lim
            ←
          
        
        ⁡
        
          Z
        
        
          /
        
        n
        
          Z
        
        ,
      
    
    {\displaystyle {\widehat {\mathbb {Z} }}=\varprojlim \mathbb {Z} /n\mathbb {Z} ,}

where the <a href="/facts/Inverse_limit/f6QecRUu">inverse limit</a> of the <a href="/facts/Quotient_ring/X2WkXyEw">quotient rings</a> 
 
 
 
 
 Z
 
 
 /
 
 n
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} /n\mathbb {Z} }
 
 runs through all <a href="/facts/Natural_number/u65og8JE">natural numbers</a> 
 
 
 
 n
 
 
 {\displaystyle n}
 
, <a href="/facts/Partial_order/qb4a3FAX">partially ordered</a> by <a href="/facts/Divisibility/DKNa2GvC">divisibility</a>. By definition, this ring is the <a href="/facts/Profinite_completion/0RX9Da9C">profinite completion</a> of the <a href="/facts/Integer/PUwwrawx">integers</a> 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
. By the <a href="/facts/Chinese_remainder_theorem/7wwFIkuV">Chinese remainder theorem</a>, 
 
 
 
 
 
 
 
 Z
 
 ^
 
 
 
 
 
 {\displaystyle {\widehat {\mathbb {Z} }}}
 
 can also be understood as the <a href="/facts/Direct_product_of_rings/wBGSEE54">direct product of rings</a>

Z
              
              ^
            
          
        
        =
        
          ∏
          
            p
          
        
        
          
            Z
          
          
            p
          
        
        ,
      
    
    {\displaystyle {\widehat {\mathbb {Z} }}=\prod _{p}\mathbb {Z} _{p},}

where the index 
 
 
 
 p
 
 
 {\displaystyle p}
 
 runs over all <a href="/facts/Prime_number/L20FLkgn">prime numbers</a>, and 
 
 
 
 
 
 Z
 
 
 p
 
 
 
 
 {\displaystyle \mathbb {Z} _{p}}
 
 is the ring of <a href="/facts/P-adic_integer/SDgDbkLF">p-adic integers</a>. This group is important because of its relation to <a href="/facts/Galois_theory/54wscJfj">Galois theory</a>, <a href="/facts/%C3%89tale_homotopy_theory/QN598o1X">étale homotopy theory</a>, and the ring of <a href="/facts/Ring_of_adeles/2LsbgeBE">adeles</a>. In addition, it provides a basic tractable example of a <a href="/facts/Profinite_group/0RX9Da9C">profinite group</a>.

Profinite integer open-in-new

Profinite integer