Locally compact skew-field

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)