View source for Locally compact skew-field

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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.

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

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

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|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  $  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|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|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====
<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>