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A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l0603101.png" /> endowed with both the algebraic structure of a [[Skew-field|skew-field]] and a locally compact topology (cf. [[Locally compact space|Locally compact space]]). It is required that the algebraic operations, that is, addition, multiplication and transitions to negative and inverse elements (the latter is defined only on the set of non-zero elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l0603102.png" />) are continuous in the given topology. Since any skew-field is locally compact with respect to the discrete topology, it is assumed that the topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l0603103.png" /> is not discrete.
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The study of locally compact skew-fields is based on the existence of a [[Haar measure|Haar measure]] on the locally compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l0603104.png" /> (the additive group of the skew-field). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l0603105.png" /> be a Haar measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l0603106.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l0603107.png" /> be a compact set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l0603108.png" /> of positive measure. Then the formula
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l0603109.png" /></td> </tr></table>
+
A set  $  K $
 +
endowed with both the algebraic structure of a [[Skew-field|skew-field]] and a locally compact topology (cf. [[Locally compact space|Locally compact space]]). It is required that the algebraic operations, that is, addition, multiplication and transitions to negative and inverse elements (the latter is defined only on the set of non-zero elements  $  K  ^ {*} = K \setminus  0 $)
 +
are continuous in the given topology. Since any skew-field is locally compact with respect to the discrete topology, it is assumed that the topology of  $  K $
 +
is not discrete.
  
defines a homomorphism (the modulus) of the multiplicative group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031010.png" /> into the multiplicative group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031011.png" /> of positive real numbers. By definition one puts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031012.png" />.
+
The study of locally compact skew-fields is based on the existence of a [[Haar measure|Haar measure]] on the locally compact group  $  K _ {+} $(
 +
the additive group of the skew-field). Let  $  \mu $
 +
be a Haar measure on  $  K _ {+} $
 +
and let  $  S \subset  K $
 +
be a compact set in  $  K $
 +
of positive measure. Then the formula
 +
 
 +
$$
 +
\mathop{\rm mod} _ {K} ( a)  = 
 +
\frac{\mu ( a S ) }{\mu ( S) }
 +
 
 +
$$
 +
 
 +
defines a homomorphism (the modulus) of the multiplicative group $  K  ^ {*} $
 +
into the multiplicative group $  \mathbf R _ {+}  ^ {*} $
 +
of positive real numbers. By definition one puts $  \mathop{\rm mod} _ {K} ( 0) = 0 $.
  
 
The  "modulus"  function satisfies the inequality
 
The  "modulus"  function satisfies the inequality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031013.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm mod} _ {K} ( a + b )  \leq  \
 +
A  \sup (  \mathop{\rm mod} _ {K} ( a) ,  \mathop{\rm mod} _ {K} ( b) )
 +
$$
  
with some constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031014.png" />. If this inequality holds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031015.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031016.png" /> is said to be non-Archimedean, or [[ultrametric]]. Otherwise <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031017.png" /> is called an Archimedean skew-field. A skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031018.png" /> is Archimedean if and only if it is connected. Any Archimedean skew-field is isomorphic to either the field of real numbers, the field of complex numbers or the skew-field of quaternions.
+
with some constant $  A > 0 $.  
 +
If this inequality holds for $  A = 1 $,  
 +
then $  K $
 +
is said to be non-Archimedean, or [[ultrametric]]. Otherwise $  K $
 +
is called an Archimedean skew-field. A skew-field $  K $
 +
is Archimedean if and only if it is connected. Any Archimedean skew-field is isomorphic to either the field of real numbers, the field of complex numbers or the skew-field of quaternions.
  
An ultrametric skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031019.png" /> is totally disconnected (cf. [[Totally-disconnected space|Totally-disconnected space]]). The  "modulus"  function determines a non-Archimedean metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031020.png" />. Any such skew-field is an extension of finite degree of either the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031021.png" /> of rational <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031022.png" />-adic numbers for some prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031023.png" /> (in the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031024.png" /> has characteristic 0) or the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031025.png" /> of [[Formal power series|formal power series]] over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031026.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031027.png" /> elements (in the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031028.png" /> has characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031029.png" />). The field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031030.png" /> (respectively, the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031031.png" />) lies in the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031032.png" />. In each of these cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031033.png" /> is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031035.png" />-skew-field, or a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031037.png" />-field.
+
An ultrametric skew-field $  K $
 +
is totally disconnected (cf. [[Totally-disconnected space|Totally-disconnected space]]). The  "modulus"  function determines a non-Archimedean metric on $  K $.  
 +
Any such skew-field is an extension of finite degree of either the field $  \mathbf Q _ {p} $
 +
of rational $  p $-
 +
adic numbers for some prime number $  p $(
 +
in the case when $  K $
 +
has characteristic 0) or the field $  \mathbf F _ {p} ( ( X) ) $
 +
of [[Formal power series|formal power series]] over the field $  \mathbf F _ {p} $
 +
of $  p $
 +
elements (in the case when $  K $
 +
has characteristic $  p $).  
 +
The field $  \mathbf Q _ {p} $(
 +
respectively, the field $  \mathbf F _ {p} ( ( X) ) $)  
 +
lies in the centre of $  K $.  
 +
In each of these cases $  K $
 +
is called a $  p $-
 +
skew-field, or a $  p $-
 +
field.
  
An ultrametric skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031038.png" /> contains a unique maximal subring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031039.png" />, defined by the condition
+
An ultrametric skew-field $  K $
 +
contains a unique maximal subring $  R $,  
 +
defined by the condition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031040.png" /></td> </tr></table>
+
$$
 +
= \{ {a \in K } : { \mathop{\rm mod} _ {K} ( a) \leq  1 } \}
 +
.
 +
$$
  
This ring is local (cf. [[Local ring|Local ring]]). Its maximal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031041.png" /> is defined by the condition
+
This ring is local (cf. [[Local ring|Local ring]]). Its maximal ideal $  P $
 +
is defined by the condition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031042.png" /></td> </tr></table>
+
$$
 +
= \{ {a \in R } : { \mathop{\rm mod} _ {K} ( a) < 1 } \}
 +
,
 +
$$
  
and all elements with modulus 1 are invertible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031043.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031044.png" /> is a [[Principal ideal|principal ideal]], and the residue field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031045.png" /> is a finite field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031046.png" />.
+
and all elements with modulus 1 are invertible in $  R $.  
 +
$  P $
 +
is a [[Principal ideal|principal ideal]], and the residue field $  R / P $
 +
is a finite field of characteristic $  p $.
  
In the case when the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031047.png" />-skew-field is not commutative, it has dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031048.png" /> over its centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031049.png" /> and ramification index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031050.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031051.png" />. Also, there is an intermediate field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031052.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031053.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031054.png" /> is an unramified extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031055.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031056.png" />, and all automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031057.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031058.png" /> are induced by inner automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060310/l06031059.png" />.
+
In the case when the $  p $-
 +
skew-field is not commutative, it has dimension $  n  ^ {2} $
 +
over its centre $  K _ {0} $
 +
and ramification index $  n $
 +
over $  K _ {0} $.  
 +
Also, there is an intermediate field $  K _ {1} $
 +
such that $  K \supset K _ {1} \supset K _ {0} $,  
 +
where $  K _ {1} $
 +
is an unramified extension of $  K _ {0} $
 +
of degree $  n $,  
 +
and all automorphisms of $  K _ {1} $
 +
over $  K _ {0} $
 +
are induced by inner automorphisms of $  K $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Weil,  "Basic number theory" , Springer  (1974)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.L. van der Waerden,  "Algebra" , '''1–2''' , Springer  (1967–1971)  (Translated from German)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.W.S. Cassels (ed.)  A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press  (1986)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L.S. Pontryagin,  "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Weil,  "Basic number theory" , Springer  (1974)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.L. van der Waerden,  "Algebra" , '''1–2''' , Springer  (1967–1971)  (Translated from German)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.W.S. Cassels (ed.)  A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press  (1986)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L.S. Pontryagin,  "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)</TD></TR></table>

Latest revision as of 22:17, 5 June 2020


A set $ K $ endowed with both the algebraic structure of a skew-field and a locally compact topology (cf. Locally compact space). It is required that the algebraic operations, that is, addition, multiplication and transitions to negative and inverse elements (the latter is defined only on the set of non-zero elements $ K ^ {*} = K \setminus 0 $) are continuous in the given topology. Since any skew-field is locally compact with respect to the discrete topology, it is assumed that the topology of $ K $ is not discrete.

The study of locally compact skew-fields is based on the existence of a Haar measure on the locally compact group $ K _ {+} $( the additive group of the skew-field). Let $ \mu $ be a Haar measure on $ K _ {+} $ and let $ S \subset K $ be a compact set in $ K $ of positive measure. Then the formula

$$ \mathop{\rm mod} _ {K} ( a) = \frac{\mu ( a S ) }{\mu ( S) } $$

defines a homomorphism (the modulus) of the multiplicative group $ K ^ {*} $ into the multiplicative group $ \mathbf R _ {+} ^ {*} $ of positive real numbers. By definition one puts $ \mathop{\rm mod} _ {K} ( 0) = 0 $.

The "modulus" function satisfies the inequality

$$ \mathop{\rm mod} _ {K} ( a + b ) \leq \ A \sup ( \mathop{\rm mod} _ {K} ( a) , \mathop{\rm mod} _ {K} ( b) ) $$

with some constant $ A > 0 $. If this inequality holds for $ A = 1 $, then $ K $ is said to be non-Archimedean, or ultrametric. Otherwise $ K $ is called an Archimedean skew-field. A skew-field $ K $ is Archimedean if and only if it is connected. Any Archimedean skew-field is isomorphic to either the field of real numbers, the field of complex numbers or the skew-field of quaternions.

An ultrametric skew-field $ K $ is totally disconnected (cf. Totally-disconnected space). The "modulus" function determines a non-Archimedean metric on $ K $. Any such skew-field is an extension of finite degree of either the field $ \mathbf Q _ {p} $ of rational $ p $- adic numbers for some prime number $ p $( in the case when $ K $ has characteristic 0) or the field $ \mathbf F _ {p} ( ( X) ) $ of formal power series over the field $ \mathbf F _ {p} $ of $ p $ elements (in the case when $ K $ has characteristic $ p $). The field $ \mathbf Q _ {p} $( respectively, the field $ \mathbf F _ {p} ( ( X) ) $) lies in the centre of $ K $. In each of these cases $ K $ is called a $ p $- skew-field, or a $ p $- field.

An ultrametric skew-field $ K $ contains a unique maximal subring $ R $, defined by the condition

$$ R = \{ {a \in K } : { \mathop{\rm mod} _ {K} ( a) \leq 1 } \} . $$

This ring is local (cf. Local ring). Its maximal ideal $ P $ is defined by the condition

$$ P = \{ {a \in R } : { \mathop{\rm mod} _ {K} ( a) < 1 } \} , $$

and all elements with modulus 1 are invertible in $ R $. $ P $ is a principal ideal, and the residue field $ R / P $ is a finite field of characteristic $ p $.

In the case when the $ p $- skew-field is not commutative, it has dimension $ n ^ {2} $ over its centre $ K _ {0} $ and ramification index $ n $ over $ K _ {0} $. Also, there is an intermediate field $ K _ {1} $ such that $ K \supset K _ {1} \supset K _ {0} $, where $ K _ {1} $ is an unramified extension of $ K _ {0} $ of degree $ n $, and all automorphisms of $ K _ {1} $ over $ K _ {0} $ are induced by inner automorphisms of $ K $.

References

[1] N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)
[2] A. Weil, "Basic number theory" , Springer (1974)
[3] B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German)
[4] J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1986)
[5] L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian)
How to Cite This Entry:
Locally compact skew-field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Locally_compact_skew-field&oldid=47689
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article