Local field

<h2 id="induced-absolute-value">Induced absolute value</h2>
Given such an absolute value on a field K, the following topology can be defined on K: for a positive real number m, define the subset Bm of K by

B
          
            m
          
        
        :=
        {
        a
        ∈
        K
        :
        
          |
        
        a
        
          |
        
        ≤
        m
        }
        .
      
    
    {\displaystyle B_{m}:=\{a\in K:|a|\leq m\}.}

Then, the b+Bm make up a <a href="/facts/Neighbourhood_basis/fMRM8Xht">neighbourhood basis</a> of b in K.
Conversely, a topological field with a non-discrete locally compact topology has an absolute value defining its topology. It can be constructed using the <a href="/facts/Haar_measure/2IcmTcuU">Haar measure</a> of the <a href="/facts/Field_(mathematics)/xAjAS4ko">additive group</a> of the field.

<h2 id="basic-features-of-non-archimedean-local-fields">Basic features of non-Archimedean local fields</h2>
For a non-Archimedean local field F (with absolute value denoted by |·|), the following objects are important:

<ul><li>its <a href="/facts/Ring_of_integers/KEzxxMbh">ring of integers</a> 
 
 
 
 
 
 O
 
 
 =
 {
 a
 ∈
 F
 :
 
 |
 
 a
 
 |
 
 ≤
 1
 }
 
 
 {\displaystyle {\mathcal {O}}=\{a\in F:|a|\leq 1\}}
 
 which is a <a href="/facts/Discrete_valuation_ring/DaHoTBSH">discrete valuation ring</a>, is the closed <a href="/facts/Unit_ball/5UeoTcug">unit ball</a> of F, and is <a href="/facts/Compact_space/d0cgXJH7">compact</a>;</li>
<li>the units in its ring of integers 
 
 
 
 
 
 
 O
 
 
 
 ×
 
 
 =
 {
 a
 ∈
 F
 :
 
 |
 
 a
 
 |
 
 =
 1
 }
 
 
 {\displaystyle {\mathcal {O}}^{\times }=\{a\in F:|a|=1\}}
 
 which forms a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> and is the <a href="/facts/Unit_sphere/5UeoTcug">unit sphere</a> of F;</li>
<li>the unique non-zero <a href="/facts/Prime_ideal/pmrGf6DA">prime ideal</a> 
 
 
 
 
 
 m
 
 
 
 
 {\displaystyle {\mathfrak {m}}}
 
 in its ring of integers which is its open unit ball 
 
 
 
 {
 a
 ∈
 F
 :
 
 |
 
 a
 
 |
 
 <
 1
 }
 
 
 {\displaystyle \{a\in F:|a|<1\}}
 
;</li>
<li>a <a href="/facts/Principal_ideal/Cga66qgM">generator</a> 
 
 
 
 ϖ
 
 
 {\displaystyle \varpi }
 
 of 
 
 
 
 
 
 m
 
 
 
 
 {\displaystyle {\mathfrak {m}}}
 
 called a <a href="/facts/Uniformizer/DaHoTBSH">uniformizer</a> of 
 
 
 
 F
 
 
 {\displaystyle F}
 
;</li>
<li>its residue field 
 
 
 
 k
 =
 
 
 O
 
 
 
 /
 
 
 
 m
 
 
 
 
 {\displaystyle k={\mathcal {O}}/{\mathfrak {m}}}
 
 which is finite (since it is compact and <a href="/facts/Discrete_space/hpvXA23u">discrete</a>).</li></ul>
Every non-zero element a of F can be written as a = ϖnu with u a unit, and n a unique integer.
The normalized valuation of F is the <a href="/facts/Surjective_function/KezhLpCb">surjective function</a> v : F → Z ∪ {∞} defined by sending a non-zero a to the unique integer n such that a = ϖnu with u a unit, and by sending 0 to ∞. If q is the <a href="/facts/Cardinality/m6VPojwT">cardinality</a> of the residue field, the absolute value on F induced by its structure as a local field is given by:<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

|
        
        a
        
          |
        
        =
        
          q
          
            −
            v
            (
            a
            )
          
        
        .
      
    
    {\displaystyle |a|=q^{-v(a)}.}

An equivalent and very important definition of a non-Archimedean local field is that it is a field that is <a href="/facts/Complete_valued_field/FXadrz5c">complete with respect to a discrete valuation</a> and whose residue field is finite.

<h3>Examples</h3>
<ol><li>The p-adic numbers: the ring of integers of Qp is the ring of p-adic integers Zp. Its prime ideal is pZp and its residue field is Z/pZ. Every non-zero element of Qp can be written as u pn where u is a unit in Zp and n is an integer, with v(u pn) = n for the normalized valuation.</li>
<li>The formal Laurent series over a finite field: the ring of integers of Fq((T)) is the ring of <a href="/facts/Formal_power_series/CNmaG0Vk">formal power series</a> Fq[[T]]. Its maximal ideal is (T) (i.e. the set of <a href="/facts/Power_series/vYh6sTps">power series</a> whose <a href="/facts/Constant_term/KRFu0aum">constant terms</a> are zero) and its residue field is Fq. Its normalized valuation is related to the (lower) degree of a formal Laurent series as follows:

v
 
 (
 
 
 ∑
 
 i
 =
 −
 m
 
 
 ∞
 
 
 
 a
 
 i
 
 
 
 T
 
 i
 
 
 
 )
 
 =
 −
 m
 
 
 {\displaystyle v\left(\sum _{i=-m}^{\infty }a_{i}T^{i}\right)=-m}
 
 (where a−m is non-zero).</li>
<li>The formal Laurent series over the complex numbers is not a local field. For example, its residue field is C[[T]]/(T) = C, which is not finite.</li></ol>
<h3>Higher unit groups</h3>
The nth higher unit group of a non-Archimedean local field F is

U
          
            (
            n
            )
          
        
        =
        1
        +
        
          
            
              m
            
          
          
            n
          
        
        =
        
          {
          
            u
            ∈
            
              
                
                  O
                
              
              
                ×
              
            
            :
            u
            ≡
            1
            
            (
            
              m
              o
              d
            
            
            
              
                
                  m
                
              
              
                n
              
            
            )
          
          }
        
      
    
    {\displaystyle U^{(n)}=1+{\mathfrak {m}}^{n}=\left\{u\in {\mathcal {O}}^{\times }:u\equiv 1\,(\mathrm {mod} \,{\mathfrak {m}}^{n})\right\}}

for n ≥ 1. The group U(1) is called the group of principal units, and any element of it is called a principal unit. The full unit group 
 
 
 
 
 
 
 O
 
 
 
 ×
 
 
 
 
 {\displaystyle {\mathcal {O}}^{\times }}
 
 is denoted U(0).
The higher unit groups form a decreasing <a href="/facts/Filtration_(mathematics)/qcvLvKRI">filtration</a> of the unit group

O
            
          
          
            ×
          
        
        ⊇
        
          U
          
            (
            1
            )
          
        
        ⊇
        
          U
          
            (
            2
            )
          
        
        ⊇
        ⋯
      
    
    {\displaystyle {\mathcal {O}}^{\times }\supseteq U^{(1)}\supseteq U^{(2)}\supseteq \cdots }

whose <a href="/facts/Quotient_group/HlwpHtjb">quotients</a> are given by

O
            
          
          
            ×
          
        
        
          /
        
        
          U
          
            (
            n
            )
          
        
        ≅
        
          
            (
            
              
                
                  O
                
              
              
                /
              
              
                
                  
                    m
                  
                
                
                  n
                
              
            
            )
          
          
            ×
          
        
        
           and 
        
        
        
          U
          
            (
            n
            )
          
        
        
          /
        
        
          U
          
            (
            n
            +
            1
            )
          
        
        ≈
        
          
            O
          
        
        
          /
        
        
          
            m
          
        
      
    
    {\displaystyle {\mathcal {O}}^{\times }/U^{(n)}\cong \left({\mathcal {O}}/{\mathfrak {m}}^{n}\right)^{\times }{\text{ and }}\,U^{(n)}/U^{(n+1)}\approx {\mathcal {O}}/{\mathfrak {m}}}

for n ≥ 1.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> (Here "
 
 
 
 ≈
 
 
 {\displaystyle \approx }
 
" means a non-canonical isomorphism.)

<h3>Structure of the unit group</h3>
The multiplicative group of non-zero elements of a non-Archimedean local field F is isomorphic to

F
          
            ×
          
        
        ≅
        (
        ϖ
        )
        ×
        
          μ
          
            q
            −
            1
          
        
        ×
        
          U
          
            (
            1
            )
          
        
      
    
    {\displaystyle F^{\times }\cong (\varpi )\times \mu _{q-1}\times U^{(1)}}

where q is the order of the residue field, and μq−1 is the group of (q−1)st roots of unity (in F). Its structure as an abelian group depends on its <a href="/facts/Characteristic_(algebra)/D2VzcQaG">characteristic</a>:

<ul><li>If F has positive characteristic p, then</li></ul>

F
          
            ×
          
        
        ≅
        
          Z
        
        ⊕
        
          Z
        
        
          /
        
        
          (
          q
          −
          1
          )
        
        ⊕
        
          
            Z
          
          
            p
          
          
            
              N
            
          
        
      
    
    {\displaystyle F^{\times }\cong \mathbf {Z} \oplus \mathbf {Z} /{(q-1)}\oplus \mathbf {Z} _{p}^{\mathbf {N} }}

where N denotes the <a href="/facts/Natural_number/u65og8JE">natural numbers</a>;
<ul><li>If F has characteristic zero (i.e. it is a finite extension of Qp of degree d), then</li></ul>

F
          
            ×
          
        
        ≅
        
          Z
        
        ⊕
        
          Z
        
        
          /
        
        (
        q
        −
        1
        )
        ⊕
        
          Z
        
        
          /
        
        
          p
          
            a
          
        
        ⊕
        
          
            Z
          
          
            p
          
          
            d
          
        
      
    
    {\displaystyle F^{\times }\cong \mathbf {Z} \oplus \mathbf {Z} /(q-1)\oplus \mathbf {Z} /p^{a}\oplus \mathbf {Z} _{p}^{d}}

where a ≥ 0 is defined so that the group of p-power roots of unity in F is 
 
 
 
 
 μ
 
 
 p
 
 a
 
 
 
 
 
 
 {\displaystyle \mu _{p^{a}}}
 
.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>
<h2 id="theory-of-local-fields">Theory of local fields</h2>
This theory includes the study of types of local fields, extensions of local fields using <a href="/facts/Hensel%27s_lemma/CPCADITS">Hensel's lemma</a>, <a href="/facts/Galois_extension/52fzNQbv">Galois extensions</a> of local fields, <a href="/facts/Ramification_group/UHXcYyGh">ramification groups</a> filtrations of <a href="/facts/Galois_group/A0p35p5c">Galois groups</a> of local fields, the behavior of the norm map on local fields, the local reciprocity homomorphism and existence theorem in <a href="/facts/Local_class_field_theory/c6xCGohI">local class field theory</a>, <a href="/facts/Local_Langlands_correspondence/zxVluK8R">local Langlands correspondence</a>, <a href="/facts/Hodge-Tate_theory/7InquNl6">Hodge-Tate theory</a> (also called <a href="/facts/P-adic_Hodge_theory/r4pehyIR">p-adic Hodge theory</a>), explicit formulas for the <a href="/facts/Hilbert_symbol/KcQ6p12r">Hilbert symbol</a> in local class field theory, see e.g.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>

<h2 id="higher-dimensional-local-fields">Higher-dimensional local fields</h2>
Main article: <a href="/facts/Higher_local_field/VMV1scRF">Higher local field</a>
A local field is sometimes called a one-dimensional local field.
A non-Archimedean local field can be viewed as the field of fractions of the completion of the <a href="/facts/Local_ring/gRTdg5Qx">local ring</a> of a one-dimensional arithmetic scheme of rank 1 at its non-singular point.
For a <a href="/facts/Non-negative_integer/u65og8JE">non-negative integer</a> n, an n-dimensional local field is a complete discrete valuation field whose residue field is an (n − 1)-dimensional local field.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> Depending on the definition of local field, a zero-dimensional local field is then either a finite field (with the definition used in this article), or a perfect field of positive characteristic.
From the geometric point of view, n-dimensional local fields with last finite residue field are naturally associated to a complete flag of subschemes of an n-dimensional arithmetic scheme.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Hensel%27s_lemma/CPCADITS">Hensel's lemma</a></li>
<li><a href="/facts/Ramification_group/UHXcYyGh">Ramification group</a></li>
<li><a href="/facts/Local_class_field_theory/c6xCGohI">Local class field theory</a></li>
<li><a href="/facts/Higher_local_field/VMV1scRF">Higher local field</a></li></ul>
<h2 id="citations">Citations</h2>

<ul><li><a href="/facts/J._W._S._Cassels/AC33eVoA">Cassels, J.W.S.</a>; <a href="/facts/Albrecht_Fr%C3%B6hlich/dSb1kUcs">Fröhlich, Albrecht</a>, eds. (1967), Algebraic Number Theory, <a href="/facts/Academic_Press/qxoKjzFy">Academic Press</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0153.07403">0153.07403</a></li>
<li><a href="/facts/Ivan_Fesenko/xfXAVdP0">Fesenko, Ivan B.</a>; Vostokov, Sergei V. (2002), Local fields and their extensions, Translations of Mathematical Monographs, vol. 121 (Second ed.), Providence, RI: <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-3259-2, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1915966">1915966</a></li>
<li><a href="/facts/James_S._Milne/8xYEQbw0">Milne, James S.</a> (2020), <a href="https://www.jmilne.org/math/CourseNotes/ant.html">Algebraic Number Theory</a> (3.08 ed.)</li>
<li><a href="/facts/J%C3%BCrgen_Neukirch/nzVc4G3F">Neukirch, Jürgen</a> (1999). Algebraic Number Theory. Vol. 322. Translated by Schappacher, Norbert. Berlin: <a href="/facts/Springer_Science%2BBusiness_Media/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-65399-8. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1697859">1697859</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0956.11021">0956.11021</a>.</li>
<li><a href="/facts/Andr%C3%A9_Weil/Q4Dd0Gr4">Weil, André</a> (1995), Basic number theory, Classics in Mathematics, Berlin, Heidelberg: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-58655-5</li>
<li><a href="/facts/Jean-Pierre_Serre/1xSr7BCh">Serre, Jean-Pierre</a> (1979), Local Fields, Graduate Texts in Mathematics, vol. 67 (First ed.), New York: Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-90424-7</li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="https://www.encyclopediaofmath.org/index.php?title=Local_field">"Local field"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li></ul>

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

<ol>
<li id="fn:1">Cassels & Fröhlich 1967, p. 129, Ch. VI, Intro.. - Cassels, J.W.S.; Fröhlich, Albrecht, eds. (1967), Algebraic Number Theory, Academic Press, Zbl 0153.07403 <a href="https://zbmath.org/?format=complete&q=an:0153.07403" target="_blank">https://zbmath.org/?format=complete&q=an:0153.07403</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Weil 1995, p. 20. - Weil, André (1995), Basic number theory, Classics in Mathematics, Berlin, Heidelberg: Springer-Verlag, ISBN 3-540-58655-5 <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Milne 2020, p. 127, Remark 7.49. - Milne, James S. (2020), Algebraic Number Theory (3.08 ed.) <a href="https://www.jmilne.org/math/CourseNotes/ant.html" target="_blank">https://www.jmilne.org/math/CourseNotes/ant.html</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Neukirch 1999, p. 134, Sec. 5. - Neukirch, Jürgen (1999). Algebraic Number Theory. Vol. 322. Translated by Schappacher, Norbert. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1697859" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1697859</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Milne 2020, p. 127, Remark 7.49. - Milne, James S. (2020), Algebraic Number Theory (3.08 ed.) <a href="https://www.jmilne.org/math/CourseNotes/ant.html" target="_blank">https://www.jmilne.org/math/CourseNotes/ant.html</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Fesenko & Vostokov 2002, Def. 1.4.6. - Fesenko, Ivan B.; Vostokov, Sergei V. (2002), Local fields and their extensions, Translations of Mathematical Monographs, vol. 121 (Second ed.), Providence, RI: American Mathematical Society, ISBN 978-0-8218-3259-2, MR 1915966 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1915966" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1915966</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Weil 1995, Ch. I, Theorem 6. - Weil, André (1995), Basic number theory, Classics in Mathematics, Berlin, Heidelberg: Springer-Verlag, ISBN 3-540-58655-5 <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Neukirch 1999, p. 122. - Neukirch, Jürgen (1999). Algebraic Number Theory. Vol. 322. Translated by Schappacher, Norbert. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1697859" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1697859</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Neukirch 1999, Theorem II.5.7. - Neukirch, Jürgen (1999). Algebraic Number Theory. Vol. 322. Translated by Schappacher, Norbert. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1697859" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1697859</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Fesenko & Vostokov 2002, Chapters 1-4, 7. - Fesenko, Ivan B.; Vostokov, Sergei V. (2002), Local fields and their extensions, Translations of Mathematical Monographs, vol. 121 (Second ed.), Providence, RI: American Mathematical Society, ISBN 978-0-8218-3259-2, MR 1915966 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1915966" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1915966</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Fesenko & Vostokov 2002, Def. 1.4.6. - Fesenko, Ivan B.; Vostokov, Sergei V. (2002), Local fields and their extensions, Translations of Mathematical Monographs, vol. 121 (Second ed.), Providence, RI: American Mathematical Society, ISBN 978-0-8218-3259-2, MR 1915966 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1915966" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1915966</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
</ol>

Local field open-in-new

Local field