Local field

Inmathematics,afieldKis called a non-Archimedeanlocal fieldif it iscompletewith respect to ametricinduced by adiscrete valuationvand if itsresidue fieldkis finite.In general, a local field is alocally compacttopological fieldwith respect to anon-discrete topology.Thereal numbersR,and thecomplex numbersC(with their standard topologies) are Archimedean local fields. Given a local field, thevaluationdefined on it can be of either of two types, each one corresponds to one of the two basic types of local fields: those in which the valuation isArchimedeanand those in which it is not. In the first case, one calls the local field anArchimedean local field,in the second case, one calls it anon-Archimedean local field.Local fields arise naturally innumber theoryas completions ofglobal fields.

While Archimedean local fields have been quite well known in mathematics for at least 250 years, the first examples of non-Archimedean local fields, the fields ofp-adic numbersfor positive prime integerp,were introduced byKurt Henselat the end of the 19th century.

Every local field isisomorphic(as a topological field) to one of the following:

• Archimedean local fields (characteristiczero): thereal numbersR,and thecomplex numbersC.
• Non-Archimedean local fields of characteristic zero:finite extensionsof thep-adic numbersQ_p(wherepis anyprime number).
• Non-Archimedean local fields of characteristicp(forpany given prime number): the field offormal Laurent seriesF_q((T)) over afinite fieldF_q,whereqis apowerofp.

In particular, of importance in number theory, classes of local fields show up as the completions ofalgebraic number fieldswith respect to their discrete valuation corresponding to one of theirmaximal ideals.Research papers in modern number theory often consider a more general notion, requiring only that the residue field beperfectof positive characteristic, not necessarily finite.This article uses the former definition.

Induced absolute value

Given such an absolute value on a fieldK,the following topology can be defined onK:for a positive real numberm,define the subsetB_mofKby

$B_{m}:=\{a\in K:|a|\leq m\}.$

Then, theb+B_mmake up aneighbourhood basisof b inK.

Conversely, a topological field with a non-discrete locally compact topology has an absolute value defining its topology. It can be constructed using theHaar measureof theadditive groupof the field.

Basic features of non-Archimedean local fields

For a non-Archimedean local fieldF(with absolute value denoted by |·|), the following objects are important:

• itsring of integers${\mathcal {O}}=\{a\in F:|a|\leq 1\}$which is adiscrete valuation ring,is the closedunit ballofF,and iscompact;
• theunitsin its ring of integers${\mathcal {O}}^{\times }=\{a\in F:|a|=1\}$which forms agroupand is theunit sphereofF;
• the unique non-zeroprime ideal${\mathfrak {m}}$in its ring of integers which is its open unit ball$\{a\in F:|a|<1\}$;
• agenerator$\varpi$of${\mathfrak {m}}$called auniformizerof$F$;
• its residue field$k={\mathcal {O}}/{\mathfrak {m}}$which is finite (since it is compact anddiscrete).

Every non-zero elementaofFcan be written asa= ϖⁿuwithua unit, andna unique integer. Thenormalized valuationofFis thesurjective functionv:F→Z∪ {∞} defined by sending a non-zeroato the unique integernsuch thata= ϖⁿuwithua unit, and by sending 0 to ∞. Ifqis thecardinalityof the residue field, the absolute value onFinduced by its structure as a local field is given by:

$|a|=q^{-v(a)}.$

An equivalent and very important definition of a non-Archimedean local field is that it is a field that iscomplete with respect to a discrete valuationand whose residue field is finite.

Examples

1. Thep-adic numbers:the ring of integers ofQ_pis the ring ofp-adic integersZ_p.Its prime ideal ispZ_pand its residue field isZ/pZ.Every non-zero element ofQ_pcan be written asupⁿwhereuis a unit inZ_pandnis an integer, thenv(upⁿ) =nfor the normalized valuation.
2. The formal Laurent series over a finite field:the ring of integers ofF_q((T)) is the ring offormal power seriesF_q[[T]]. Its maximal ideal is (T) (i.e. thepower serieswhoseconstant termis zero) and its residue field isF_q.Its normalized valuation is related to the (lower) degree of a formal Laurent series as follows:

   $v\left(\sum _{i=-m}^{\infty }a_{i}T^{i}\right)=-m$(wherea_−mis non-zero).

3. The formal Laurent series over the complex numbers isnota local field. For example, its residue field isC[[T]]/(T) =C,which is not finite.

Higher unit groups

Then^thhigher unit groupof a non-Archimedean local fieldFis

$U^{(n)}=1+{\mathfrak {m}}^{n}=\left\{u\in {\mathcal {O}}^{\times }:u\equiv 1\,(\mathrm {mod} \,{\mathfrak {m}}^{n})\right\}$

forn≥ 1. The groupU⁽¹⁾is called thegroup of principal units,and any element of it is called aprincipal unit.The full unit group${\mathcal {O}}^{\times }$is denotedU⁽⁰⁾.

The higher unit groups form a decreasingfiltrationof the unit group

${\mathcal {O}}^{\times }\supseteq U^{(1)}\supseteq U^{(2)}\supseteq \cdots$

whosequotientsare given by

${\mathcal {O}}^{\times }/U^{(n)}\cong \left({\mathcal {O}}/{\mathfrak {m}}^{n}\right)^{\times }{\text{ and }}\,U^{(n)}/U^{(n+1)}\approx {\mathcal {O}}/{\mathfrak {m}}$

forn≥ 1.(Here "$\approx$"means a non-canonical isomorphism.)

Structure of the unit group

The multiplicative group of non-zero elements of a non-Archimedean local fieldFis isomorphic to

$F^{\times }\cong (\varpi )\times \mu _{q-1}\times U^{(1)}$

whereqis the order of the residue field, and μ_q−1is the group of (q−1)st roots of unity (inF). Its structure as an abelian group depends on itscharacteristic:

• IfFhas positive characteristicp,then

$F^{\times }\cong \mathbf {Z} \oplus \mathbf {Z} /{(q-1)}\oplus \mathbf {Z} _{p}^{\mathbf {N} }$

whereNdenotes thenatural numbers;

• IfFhas characteristic zero (i.e. it is a finite extension ofQ_pof degreed), then

$F^{\times }\cong \mathbf {Z} \oplus \mathbf {Z} /(q-1)\oplus \mathbf {Z} /p^{a}\oplus \mathbf {Z} _{p}^{d}$

wherea≥ 0 is defined so that the group ofp-power roots of unity inFis$\mu _{p^{a}}$.

Theory of local fields

This theory includes the study of types of local fields, extensions of local fields usingHensel's lemma,Galois extensionsof local fields,ramification groupsfiltrations ofGalois groupsof local fields, the behavior of the norm map on local fields, the local reciprocity homomorphism and existence theorem inlocal class field theory,local Langlands correspondence,Hodge-Tate theory(also calledp-adic Hodge theory), explicit formulas for theHilbert symbolin local class field theory, see e.g.

Higher-dimensional local fields

Main article:Higher local field

A local field is sometimes called aone-dimensional local field.

A non-Archimedean local field can be viewed as the field of fractions of the completion of thelocal ringof a one-dimensional arithmetic scheme of rank 1 at its non-singular point.

For anon-negative integern,ann-dimensional local field is a complete discrete valuation field whose residue field is an (n− 1)-dimensional local field.Depending on the definition of local field, azero-dimensional local fieldis then either a finite field (with the definition used in this article), or a perfect field of positive characteristic.

From the geometric point of view,n-dimensional local fields with last finite residue field are naturally associated to a complete flag of subschemes of ann-dimensional arithmetic scheme.

See also

• Hensel's lemma
• Ramification group
• Local class field theory
• Higher local field

References

• Cassels, J.W.S.;Fröhlich, Albrecht,eds. (1967),Algebraic Number Theory,Academic Press,Zbl0153.07403
• Fesenko, Ivan B.;Vostokov, Sergei V. (2002),Local fields and their extensions,Translations of Mathematical Monographs, vol. 121 (Second ed.), Providence, RI:American Mathematical Society,ISBN978-0-8218-3259-2,MR1915966
• Milne, James S.(2020),Algebraic Number Theory(3.08 ed.)
• Neukirch, Jürgen(1999).Algebraic Number Theory.Vol. 322. Translated by Schappacher, Norbert. Berlin:Springer-Verlag.ISBN978-3-540-65399-8.MR1697859.Zbl0956.11021.
• Weil, André(1995),Basic number theory,Classics in Mathematics, Berlin, Heidelberg:Springer-Verlag,ISBN3-540-58655-5
• Serre, Jean-Pierre(1979),Local Fields,Graduate Texts in Mathematics, vol. 67 (First ed.), New York: Springer-Verlag,ISBN0-387-90424-7

External links

• "Local field",Encyclopedia of Mathematics,EMS Press,2001 [1994]