Local body

In algebra and number theory, a local field is a topological field whose underlying topology is locally compact and not discrete . The topology of such a body can always be described by an amount . There are two fundamentally different types of local bodies: Archimedean local bodies and non-Archimedean local bodies .

Local bodies can be fully classified:

• Archimedean local bodies are always isomorphic to or . ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$
• Non-Archimedean local fields of the characteristic are always isomorphic to a finite field extension of the - adic numbers (for a prime number ). ${\displaystyle 0}$ ${\displaystyle p}$ ${\displaystyle \mathbb {Q} _{p}}$ ${\displaystyle p}$
• Non-Archimedean local fields of the characteristic are always isomorphic to the field of the formal Laurent series, being a finite field of the characteristic and a formal variable. ${\displaystyle p>0}$ ${\displaystyle F((t))}$ ${\displaystyle F}$ ${\displaystyle p}$ ${\displaystyle t}$

Non-Archimedean local fields can also be characterized equivalently as fields that are complete with regard to a non-trivial discrete evaluation and have a finite residue class field . Such local fields appear in algebraic number theory as completions of global fields .

Usage and terminology

In number theory one is interested in solving equations over the field of rational numbers , a global field that has the characteristic . According to Ostrowski's theorem , there are two types of absolute value functions, one being Archimedean (with respect to which the rational numbers can be completed to the real numbers) and a family of non-Archimedean evaluations (with regard to which they can be completed to the p-adic numbers) . The associated local fields are the real and p-adic numbers. According to the Hasse principle ( local-global principle according to Helmut Hasse ) one can sometimes deduce from the solvability over local fields on the solvability in the global field of the rational numbers, e.g. for non-degenerate quadratic forms. With the help of local fields, the local class field theory is formulated, also justified by Hasse, and used by Claude Chevalley to build the global class field theory without recourse to methods of analytical number theory. The representation of the local class field theory with the help of group cohomology has been a standard approach since the seminar of Emil Artin and John T. Tate and is presented, for example, in the book by Serre Local Fields . ${\displaystyle \mathbb {Q} }$ ${\displaystyle 0}$

As with the terms " local ring " and " localization " in algebra, the term local has its origin in the analogy of the number field case with the case of a function field over a complex algebraic curve ( Riemann surface ), where "local" is the behavior of the Functions in the vicinity of a point describes and “globally” the possibility of combining the function defined in a local vicinity of points, for example via a power series, over the entire Riemann surface to form a global function.

Properties of non-Archimedean local bodies

Given a non-Archimedean local field with magnitude | · |. Then the following objects are important: ${\displaystyle K}$

• The evaluation ring : A local main ideal ring , which at the same time represents the closed unit sphere in . ${\displaystyle {\mathcal {O}}=\{a\in K:|a|\leq 1\}}$ ${\displaystyle K}$
• the maximum ideal of : the open unit sphere of . ${\displaystyle {\mathfrak {m}}=\{a\in K:|a|<1\}}$ ${\displaystyle {\mathcal {O}}}$ ${\displaystyle K}$
• the remainder class field , which must be finite as the quotient of the compact group after an open group (because it is compact and discrete) ${\displaystyle k={\mathcal {O}}/{\mathfrak {m}}}$ ${\displaystyle {\mathcal {O}}}$ ${\displaystyle {\mathfrak {m}}}$

Examples

1. The -adic numbers : The evaluation ring is , the ring of the whole -adic numbers. The maximum ideal contained therein is , that is , the main ideal that is generated by. The remainder class field ${\displaystyle p}$ ${\displaystyle K=\mathbb {Q} _{p}}$ ${\displaystyle {\mathcal {O}}=\mathbb {Z} _{p}}$ ${\displaystyle p}$ ${\displaystyle {\mathfrak {m}}=p\mathbb {Z} _{p}}$ ${\displaystyle p}$ ${\displaystyle k={\mathcal {O}}/{\mathfrak {m}}=\mathbb {Z} _{p}/(p\mathbb {Z} _{p})\cong \mathbb {Z} /p\mathbb {Z} }$
2. The formal Laurent series of a formal variable over a finite field : The evaluation ring is , the ring of the formal power series in over . The maximum ideal contained therein is , i.e. the set of all power series with a constant term . The remainder class field ${\displaystyle F((t))}$ ${\displaystyle t}$ ${\displaystyle F}$ ${\displaystyle {\mathcal {O}}=F[[t]]}$ ${\displaystyle t}$ ${\displaystyle F}$ ${\displaystyle {\mathfrak {m}}=tF[[t]]}$ ${\displaystyle 0}$ ${\displaystyle k={\mathcal {O}}/{\mathfrak {m}}=F[[t]]/tF[[t]]\cong F}$
3. The formal Laurent series above are not a local field since their remainder class field is isomorphic to , which is not finite. ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {C} }$

Generalizations

There is a generalization of the local bodies by the so-called higher local bodies. For an n-local field is a field that is complete with respect to a discrete valuation and whose remainder class field is an (n-1) -local field. The 1-local bodies are the usual local bodies. For example, or are 2-local bodies. ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle \mathbb {Q} _{p}((t))}$ ${\displaystyle \mathbb {F} _{p}((t_{1}))((t_{2}))}$

literature

• Jean-Pierre Serre : Local Fields (= Graduate Texts in Mathematics. Vol. 67). Springer, New York, NY et al. 1979, ISBN 0-387-90424-7 .
• Jürgen Neukirch : Algebraic number theory. Springer, Berlin et al. 1992, ISBN 3-540-54273-6 .

Web links

• Tim Browning, Local Fields, Lecture Notes Univ. Warwick, pdf
• Danilov, Local field, Encyclopedia of Mathematics (in the definition there only bodies with a discrete evaluation function are considered)

Individual evidence

1. ^ André Weil Basic number theory , Springer-Verlag 1995; Page 20
2. ^ IB Fesenko , SV Vostokov Local fields and their extensions. A constructive approach , American Mathematical Society 1993, 2nd edition 2002
3. ^ Fesenko, Masato Kurihara (editor) Invitation to higher local fields , Geometry and Topology Monographs 3, University of Warwick 2000, online