View source for Locally compact skew-field

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.

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

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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  "modulus"  function satisfies the inequality

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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.

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 <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

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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

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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" />.

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" />.

====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>