SherbyNet - /page/wikibook/entry.mu

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Local field

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In <a href="Mathematics" title="Mathematics">mathematics</a>, a local field is a certain type of <a href="Topological_field" class="mw-redirect" title="Topological field">topological field</a>: by definition, a local field is a <a href="Locally_compact" class="mw-redirect" title="Locally compact">locally compact</a> <a href="Hausdorff_space" title="Hausdorff space">Hausdorff</a> non-<a href="Discrete_space" title="Discrete space">discrete</a> <a href="Topological_field" class="mw-redirect" title="Topological field">topological field</a>.<a href="#cite_note-FOOTNOTEWeil199520-1">[1]</a> Local fields find many applications in <a href="Algebraic_number_theory" title="Algebraic number theory">algebraic number theory</a>, where they arise naturally as <a href="Complete_metric_space#Completion" title="Complete metric space">completions</a> of <a href="Global_field" title="Global field">global fields</a>.<a href="#cite_note-FOOTNOTENeukirch1999134Sec._5-2">[2]</a> Further, tools like <a href="Integral" title="Integral">integration</a> and <a href="Fourier_analysis" title="Fourier analysis">Fourier analysis</a> are available for functions defined on local fields.

Given a local field, an <a href="Absolute_value_(algebra)" title="Absolute value (algebra)">absolute value</a> can be defined on it which gives rise to a <a href="Complete_metric_space" title="Complete metric space">complete metric</a> that generates its topology. There are two basic types of local field: those called Archimedean local fields in which the absolute value is <a href="Archimedean_property#Definition_for_normed_fields" title="Archimedean property">Archimedean</a>, and those called non-Archimedean local fields in which it is not. The non-Archimedean local fields can also be defined as those fields which are complete with respect to a <a href="Metric_space" title="Metric space">metric</a> induced by a <a href="Discrete_valuation" title="Discrete valuation">discrete valuation</a> v whose <a href="Residue_field" title="Residue field">residue field</a> is finite.<a href="#cite_note-FOOTNOTECasselsFröhlich1967129Ch._VI,_Intro.-3">[3]</a>

Every local field is <a href="Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> (as a topological field) to one of the following:<a href="#cite_note-FOOTNOTEMilne2020127Remark_7.49-4">[4]</a>

<ul><li>Archimedean local fields (<a href="Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a> zero): the <a href="Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a> R, and the <a href="Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a> C.</li>

<li>Non-Archimedean local fields of characteristic zero: <a href="Finite_extension" class="mw-redirect" title="Finite extension">finite extensions</a> of the <a href="P-adic_number" title="P-adic number">p-adic numbers</a> Qp (where p is any <a href="Prime_number" title="Prime number">prime number</a>).</li>

<li>Non-Archimedean local fields of characteristic p (for p any given prime number): the field Fq((T)) of <a href="Formal_Laurent_series" class="mw-redirect" title="Formal Laurent series">formal Laurent series</a> in the variable T over a <a href="Finite_field" title="Finite field">finite field</a> Fq, where q is a <a href="Exponentiation" title="Exponentiation">power</a> of p.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Module,_absolute_value,_metric">Module, absolute value, metric</h2></div>

Given a local field F, a "module function" on F can be defined as follows. First, consider the <a href="Field_(mathematics)#Related_algebraic_structures" title="Field (mathematics)">additive group</a> of the field. As a locally compact <a href="Topological_group" title="Topological group">topological group</a>, it has a unique (up to positive scalar multiple) <a href="Haar_measure" title="Haar measure">Haar measure</a> μ. The module of an element a of F is defined so as to measure the change in size of a set after multiplying it by a. Specifically, define modK : F → R by<a href="#cite_note-FOOTNOTEWeil19954-5">[5]</a>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {mod} _{K}(a):={\frac {\mu (aX)}{\mu (X)}}}</annotation>

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</math><img src="./dd5f2ce6d1de13aed58fe07cc4a950382a7d881b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.125ex; height:6.509ex;" alt="{\displaystyle \operatorname {mod} _{K}(a):={\frac {\mu (aX)}{\mu (X)}}}" loading="lazy"></dd></dl>

for any <a href="Measurable" class="mw-redirect" title="Measurable">measurable</a> subset X of F (with 0 < μ(X) < ∞). This module does not depend on X nor on the choice of Haar measure μ (since the same scalar multiple ambiguity will occur in both the numerator and the denominator). The function modK is continuous and satisfies

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {mod} _{K}(ab)=\operatorname {mod} _{K}(a)\operatorname {mod} _{K}(b)}</annotation>

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</math><img src="./320ad24057dfa8b44ddc56c1da93decea69df3b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.62ex; height:2.843ex;" alt="{\displaystyle \operatorname {mod} _{K}(ab)=\operatorname {mod} _{K}(a)\operatorname {mod} _{K}(b)}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {mod} _{K}(a+b)\leq A\sup \left(\operatorname {mod} _{K}(a),\operatorname {mod} _{K}(b)\right)}</annotation>

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</math><img src="./3e0c0f61503e1a85347a4e16b95c85580bdf1cf8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.935ex; height:2.843ex;" alt="{\displaystyle \operatorname {mod} _{K}(a+b)\leq A\sup \left(\operatorname {mod} _{K}(a),\operatorname {mod} _{K}(b)\right)}" loading="lazy"></dd></dl>

for some constant A that only depends on F.

Using modK, one may then define an absolute value |.| on F that induces a <a href="Metric_space" title="Metric space">metric</a> on F (via the standard d(x,y) = |x-y|), such that F is complete with respect to this metric, and the metric induces the given topology on F.

<div class="mw-heading mw-heading2"><h2 id="Basic_features_of_non-Archimedean_local_fields">Basic features of non-Archimedean local fields</h2></div>

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

<ul><li>its <a href="Ring_of_integers" title="Ring of integers">ring of integers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}=\{a\in F:|a|\leq 1\}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}=\{a\in F:|a|\leq 1\}}</annotation>

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</math><img src="./e2c157b4e84720e52a717fc04ca1e5c8b7a89cde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.806ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}=\{a\in F:|a|\leq 1\}}" loading="lazy"> which is a <a href="Discrete_valuation_ring" title="Discrete valuation ring">discrete valuation ring</a>, is the closed <a href="Unit_ball" class="mw-redirect" title="Unit ball">unit ball</a> of F, and is <a href="Compact_space" title="Compact space">compact</a>;</li>

<li>the units in its ring of integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}^{\times }=\{a\in F:|a|=1\}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}^{\times }=\{a\in F:|a|=1\}}</annotation>

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</math><img src="./3e3e4eb1ab2de2c46d940bc64eb5b446ec179539.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.317ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}^{\times }=\{a\in F:|a|=1\}}" loading="lazy"> which forms a <a href="Group_(mathematics)" title="Group (mathematics)">group</a> and is the <a href="Unit_sphere" title="Unit sphere">unit sphere</a> of F;</li>

<li>the unique non-zero <a href="Prime_ideal" title="Prime ideal">prime ideal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation>

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</math><img src="./adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy"> in its ring of integers which is its open unit ball <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{a\in F:|a|<1\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{a\in F:|a|<1\}}</annotation>

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</math><img src="./d06e289fc17c15d7081e7addb55c8e74893a0186.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.858ex; height:2.843ex;" alt="{\displaystyle \{a\in F:|a|<1\}}" loading="lazy">;</li>

<li>a <a href="Principal_ideal" title="Principal ideal">generator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varpi }">

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<annotation encoding="application/x-tex">{\displaystyle \varpi }</annotation>

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</math><img src="./e50d258418b5fa150a86b58f8d5eb40613e3ebf7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:1.676ex;" alt="{\displaystyle \varpi }" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation>

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</math><img src="./adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy"> called a <a href="Uniformizer" class="mw-redirect" title="Uniformizer">uniformizer</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

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<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

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</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy">;</li>

<li>its residue field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k={\mathcal {O}}/{\mathfrak {m}}}">

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<annotation encoding="application/x-tex">{\displaystyle k={\mathcal {O}}/{\mathfrak {m}}}</annotation>

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</math><img src="./6dca0d5c3a19382a8f108b61cc95200674e29e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.105ex; height:2.843ex;" alt="{\displaystyle k={\mathcal {O}}/{\mathfrak {m}}}" loading="lazy"> which is finite (since it is compact and <a href="Discrete_space" title="Discrete space">discrete</a>).</li></ul>

Every non-zero element a of F can be written as a = ϖnu with u a unit in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}^{\times }}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}^{\times }}</annotation>

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</math><img src="./e62834984c784c857217d5a44a793b333dd75e42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.361ex; height:2.343ex;" alt="{\displaystyle {\mathcal {O}}^{\times }}" loading="lazy">, and n a unique integer.

The normalized valuation of F is the <a href="Surjective_function" title="Surjective function">surjective function</a> v : F → Z ∪ {∞} defined by sending a non-zero a to the unique integer n such that a = ϖnu with u a unit, and by sending 0 to ∞. If q is the <a href="Cardinality" title="Cardinality">cardinality</a> of the residue field, the absolute value on F induced by its structure as a local field is given by:<a href="#cite_note-FOOTNOTEWeil1995Ch._I,_Theorem_6-6">[6]</a>

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<annotation encoding="application/x-tex">{\displaystyle |a|=q^{-v(a)}.}</annotation>

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</math><img src="./b87fd92fb1d957163f6c2e2b0137476b1796a491.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.805ex; height:3.343ex;" alt="{\displaystyle |a|=q^{-v(a)}.}" loading="lazy"></dd></dl>

An equivalent and very important definition of a non-Archimedean local field is that it is a field that is <a href="Complete_valued_field" class="mw-redirect" title="Complete valued field">complete with respect to a discrete valuation</a> and whose residue field is finite.

<div class="mw-heading mw-heading3"><h3 id="Examples">Examples</h3></div>

<ol><li>The p-adic numbers: the ring of integers of Qp is the ring of p-adic integers Zp. Its prime ideal is pZp and its residue field is Z/pZ. Every non-zero element of Qp can be written as u pn where u is a unit in Zp and n is an integer, with v(u pn) = n for the normalized valuation.</li>

<li>The formal Laurent series over a finite field: the ring of integers of Fq((T)) is the ring of <a href="Formal_power_series" title="Formal power series">formal power series</a> Fq[[T]]. Its maximal ideal is (T) (i.e. the set of <a href="Power_series" title="Power series">power series</a> whose <a href="Constant_term" title="Constant term">constant terms</a> are zero) and its residue field is Fq. Its normalized valuation is related to the (lower) degree of a formal Laurent series as follows:

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<annotation encoding="application/x-tex">{\displaystyle v{\biggl (}\sum _{i=-m}^{\infty }a_{i}T^{i}{\biggr )}=-m}</annotation>

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</math><img src="./27424399e5f3672ee3a670035a2bba044232b038.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.386ex; height:7.009ex;" alt="{\displaystyle v{\biggl (}\sum _{i=-m}^{\infty }a_{i}T^{i}{\biggr )}=-m}" loading="lazy"> (where a−m is non-zero).</dd></dl></dd></dl></li>

<li>The field C((T)) of formal Laurent series over the complex numbers is not a local field. Its residue field is C[[T]]/(T) = C, which is not finite.</li></ol>

<div class="mw-heading mw-heading3"><h3 id="Higher_unit_groups">Higher unit groups</h3></div>

The nth higher unit group of a non-Archimedean local field F is

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<annotation encoding="application/x-tex">{\displaystyle U^{(n)}=1+{\mathfrak {m}}^{n}=\left\{u\in {\mathcal {O}}^{\times }:u\equiv 1\,(\mathrm {mod} \,{\mathfrak {m}}^{n})\right\}}</annotation>

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</math><img src="./d743632c7a05371f12ebdc941f9644f209fae01a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.899ex; height:3.343ex;" alt="{\displaystyle U^{(n)}=1+{\mathfrak {m}}^{n}=\left\{u\in {\mathcal {O}}^{\times }:u\equiv 1\,(\mathrm {mod} \,{\mathfrak {m}}^{n})\right\}}" loading="lazy"></dd></dl>

for n ≥ 1. The group U(1) is called the group of principal units, and any element of it is called a principal unit. The full unit group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}^{\times }}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}^{\times }}</annotation>

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</math><img src="./e62834984c784c857217d5a44a793b333dd75e42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.361ex; height:2.343ex;" alt="{\displaystyle {\mathcal {O}}^{\times }}" loading="lazy"> is denoted U(0).

The higher unit groups form a decreasing <a href="Filtration_(mathematics)" title="Filtration (mathematics)">filtration</a> of the unit group

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}^{\times }\supseteq U^{(1)}\supseteq U^{(2)}\supseteq \cdots }</annotation>

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</math><img src="./66ca01d758040ab9bc6f283cde584824dc695d72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:23.729ex; height:3.009ex;" alt="{\displaystyle {\mathcal {O}}^{\times }\supseteq U^{(1)}\supseteq U^{(2)}\supseteq \cdots }" loading="lazy"></dd></dl>

whose <a href="Quotient_group" title="Quotient group">quotients</a> are given by

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<annotation encoding="application/x-tex">{\displaystyle {\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}}}</annotation>

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</math><img src="./e81f23daa49ad9ff950a7685dfa6893c4aa50aa0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.425ex; height:3.343ex;" alt="{\displaystyle {\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}}}" loading="lazy"></dd></dl>

for n ≥ 1.<a href="#cite_note-FOOTNOTENeukirch1999122-7">[7]</a> (Here "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx }">

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<annotation encoding="application/x-tex">{\displaystyle \approx }</annotation>

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</math><img src="./6f58f4c2b73283ce8a5ad28fb3746f2a8c998789.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \approx }" loading="lazy">" means a non-canonical isomorphism.)

<div class="mw-heading mw-heading3"><h3 id="Structure_of_the_unit_group">Structure of the unit group</h3></div>

The multiplicative group of non-zero elements of a non-Archimedean local field F is isomorphic to

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<annotation encoding="application/x-tex">{\displaystyle F^{\times }\cong (\varpi )\times \mu _{q-1}\times U^{(1)}}</annotation>

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</math><img src="./56f9ce601007983f5400e1e38a87946e8d419b71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.504ex; height:3.509ex;" alt="{\displaystyle F^{\times }\cong (\varpi )\times \mu _{q-1}\times U^{(1)}}" loading="lazy"></dd></dl>

where q is the order of the residue field, and μq−1 is the group of (q−1)st roots of unity (in F). Its structure as an abelian group depends on its <a href="Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a>:

<ul><li>If F has positive characteristic p, then</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle F^{\times }\cong \mathbf {Z} \oplus \mathbf {Z} /{(q-1)}\oplus \mathbf {Z} _{p}^{\mathbf {N} }}</annotation>

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</math><img src="./c158c10d0236b5de166e3e734716fd1f52ee5a9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:26.762ex; height:3.509ex;" alt="{\displaystyle F^{\times }\cong \mathbf {Z} \oplus \mathbf {Z} /{(q-1)}\oplus \mathbf {Z} _{p}^{\mathbf {N} }}" loading="lazy"></dd></dl></dd>

<dd>where N denotes the <a href="Natural_number" title="Natural number">natural numbers</a>;</dd></dl>

<ul><li>If F has characteristic zero (i.e. it is a finite extension of Qp of degree d), then</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle F^{\times }\cong \mathbf {Z} \oplus \mathbf {Z} /(q-1)\oplus \mathbf {Z} /p^{a}\oplus \mathbf {Z} _{p}^{d}}</annotation>

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</math><img src="./a9228cbf7ba16b321464491e7644e6e7d2dd4b38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:34.051ex; height:3.509ex;" alt="{\displaystyle F^{\times }\cong \mathbf {Z} \oplus \mathbf {Z} /(q-1)\oplus \mathbf {Z} /p^{a}\oplus \mathbf {Z} _{p}^{d}}" loading="lazy"></dd></dl></dd>

<dd>where a ≥ 0 is defined so that the group of p-power roots of unity in F is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{p^{a}}}">

<msub>

<msup>

</mrow>

</msup>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu _{p^{a}}}</annotation>

</semantics>

</math><img src="./f5132fd10e56ea6bb9550f29ba05845c338e8bed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.331ex; height:2.343ex;" alt="{\displaystyle \mu _{p^{a}}}" loading="lazy">.<a href="#cite_note-FOOTNOTENeukirch1999Theorem_II.5.7-8">[8]</a></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Theory_of_local_fields">Theory of local fields</h2></div>

This theory includes the study of types of local fields, extensions of local fields using <a href="Hensel's_lemma" title="Hensel's lemma">Hensel's lemma</a>, <a href="Galois_extension" title="Galois extension">Galois extensions</a> of local fields, <a href="Ramification_group" title="Ramification group">ramification groups</a> filtrations of <a href="Galois_group" title="Galois group">Galois groups</a> of local fields, the behavior of the norm map on local fields, the local reciprocity homomorphism and existence theorem in <a href="Local_class_field_theory" title="Local class field theory">local class field theory</a>, <a href="Local_Langlands_correspondence" class="mw-redirect" title="Local Langlands correspondence">local Langlands correspondence</a>, <a href="Hodge-Tate_theory" class="mw-redirect" title="Hodge-Tate theory">Hodge-Tate theory</a> (also called <a href="P-adic_Hodge_theory" title="P-adic Hodge theory">p-adic Hodge theory</a>), explicit formulas for the <a href="Hilbert_symbol" title="Hilbert symbol">Hilbert symbol</a> in local class field theory, see e.g.<a href="#cite_note-FOOTNOTEFesenkoVostokov2002Chapters_1-4,_7-9">[9]</a>

<div class="mw-heading mw-heading2"><h2 id="Variant_definitions">Variant definitions</h2></div>

The definition for "local field" adopted in this article, as a locally compact Hausdorff non-discrete topological field, is common today. Some authors however reserve the term "local field" for what we have called "non-Archimedian local field".

Research papers in modern number theory often consider a more general notion of non-Archimedean local field, requiring only that they be complete with respect to a <a href="Discrete_valuation" title="Discrete valuation">discrete valuation</a> and that the residue field be <a href="Perfect_field" title="Perfect field">perfect</a> of positive characteristic, not necessarily finite.<a href="#cite_note-FOOTNOTEFesenkoVostokov2002Def._1.4.6-10">[10]</a>

Serre in his 1962 book <a href="Local_Fields" title="Local Fields">Local Fields</a> defined "local fields" as fields that are complete with respect to a discrete valuation, without any restriction on the residue field, leading to a notion that is more general still.

<div class="mw-heading mw-heading2"><h2 id="Higher-dimensional_local_fields">Higher-dimensional local fields</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Higher_local_field" title="Higher local field">Higher local field</a></div>

A local field is sometimes called a one-dimensional local field.

A non-Archimedean local field can be viewed as the field of fractions of the completion of the <a href="Local_ring" title="Local ring">local ring</a> of a one-dimensional arithmetic scheme of rank 1 at its non-singular point.

For a <a href="Non-negative_integer" class="mw-redirect" title="Non-negative integer">non-negative integer</a> n, an n-dimensional local field is a complete discrete valuation field whose residue field is an (n − 1)-dimensional local field.<a href="#cite_note-FOOTNOTEFesenkoVostokov2002Def._1.4.6-10">[10]</a> Depending on the definition of local field, a zero-dimensional local field is 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 <a href="Flag_(mathematics)" class="mw-redirect mw-disambig" title="Flag (mathematics)">flag</a> of subschemes of an n-dimensional <a href="Arithmetic_scheme" class="mw-redirect" title="Arithmetic scheme">arithmetic scheme</a>.

<ul><li><a href="Hensel's_lemma" title="Hensel's lemma">Hensel's lemma</a></li>

<li><a href="Ramification_group" title="Ramification group">Ramification group</a></li>

<li><a href="Local_class_field_theory" title="Local class field theory">Local class field theory</a></li>

<li><a href="Higher_local_field" title="Higher local field">Higher local field</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2></div>

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<li id="cite_note-FOOTNOTEWeil199520-1"><a href="#cite_ref-FOOTNOTEWeil199520_1-0">^</a> <a href="#CITEREFWeil1995">Weil 1995</a>, p. 20.

</li>

<li id="cite_note-FOOTNOTENeukirch1999134Sec._5-2"><a href="#cite_ref-FOOTNOTENeukirch1999134Sec._5_2-0">^</a> <a href="#CITEREFNeukirch1999">Neukirch 1999</a>, p. 134, Sec. 5.

</li>

<li id="cite_note-FOOTNOTECasselsFröhlich1967129Ch._VI,_Intro.-3"><a href="#cite_ref-FOOTNOTECasselsFröhlich1967129Ch._VI,_Intro._3-0">^</a> <a href="#CITEREFCasselsFröhlich1967">Cassels & Fröhlich 1967</a>, p. 129, Ch. VI, Intro..

</li>

<li id="cite_note-FOOTNOTEMilne2020127Remark_7.49-4"><a href="#cite_ref-FOOTNOTEMilne2020127Remark_7.49_4-0">^</a> <a href="#CITEREFMilne2020">Milne 2020</a>, p. 127, Remark 7.49.

</li>

<li id="cite_note-FOOTNOTEWeil19954-5"><a href="#cite_ref-FOOTNOTEWeil19954_5-0">^</a> <a href="#CITEREFWeil1995">Weil 1995</a>, p. 4.

</li>

<li id="cite_note-FOOTNOTEWeil1995Ch._I,_Theorem_6-6"><a href="#cite_ref-FOOTNOTEWeil1995Ch._I,_Theorem_6_6-0">^</a> <a href="#CITEREFWeil1995">Weil 1995</a>, Ch. I, Theorem 6.

</li>

<li id="cite_note-FOOTNOTENeukirch1999122-7"><a href="#cite_ref-FOOTNOTENeukirch1999122_7-0">^</a> <a href="#CITEREFNeukirch1999">Neukirch 1999</a>, p. 122.

</li>

<li id="cite_note-FOOTNOTENeukirch1999Theorem_II.5.7-8"><a href="#cite_ref-FOOTNOTENeukirch1999Theorem_II.5.7_8-0">^</a> <a href="#CITEREFNeukirch1999">Neukirch 1999</a>, Theorem II.5.7.

</li>

<li id="cite_note-FOOTNOTEFesenkoVostokov2002Chapters_1-4,_7-9"><a href="#cite_ref-FOOTNOTEFesenkoVostokov2002Chapters_1-4,_7_9-0">^</a> <a href="#CITEREFFesenkoVostokov2002">Fesenko & Vostokov 2002</a>, Chapters 1-4, 7.

</li>

<li id="cite_note-FOOTNOTEFesenkoVostokov2002Def._1.4.6-10">^ <a href="#cite_ref-FOOTNOTEFesenkoVostokov2002Def._1.4.6_10-0">a</a> <a href="#cite_ref-FOOTNOTEFesenkoVostokov2002Def._1.4.6_10-1">b</a> <a href="#CITEREFFesenkoVostokov2002">Fesenko & Vostokov 2002</a>, Def. 1.4.6.

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFCasselsFröhlich1967" class="citation cs2"><a href="J._W._S._Cassels" title="J. W. S. Cassels">Cassels, J.W.S.</a>; <a href="Albrecht_Fr%C3%B6hlich" title="Albrecht Fröhlich">Fröhlich, Albrecht</a>, eds. (1967), Algebraic Number Theory, <a href="Academic_Press" title="Academic Press">Academic Press</a>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0153.07403">0153.07403</a></cite></li>

<li><cite id="CITEREFFesenkoVostokov2002" class="citation cs2"><a href="Ivan_Fesenko" title="Ivan Fesenko">Fesenko, Ivan B.</a>; Vostokov, Sergei V. (2002), Local fields and their extensions, Translations of Mathematical Monographs, vol. 121 (Second ed.), Providence, RI: <a href="American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8218-3259-2</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1915966">1915966</a></cite></li>

<li><cite id="CITEREFMilne2020" class="citation cs2"><a href="James_S._Milne" class="mw-redirect" title="James S. Milne">Milne, James S.</a> (2020), <a rel="nofollow" class="external text" href="https://www.jmilne.org/math/CourseNotes/ant.html">Algebraic Number Theory</a> (3.08 ed.)</cite></li>

<li><cite id="CITEREFNeukirch1999" class="citation book cs1"><a href="J%C3%BCrgen_Neukirch" title="Jürgen Neukirch">Neukirch, Jürgen</a> (1999). Algebraic Number Theory. Vol. 322. Translated by Schappacher, Norbert. Berlin: <a href="Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer-Verlag</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-65399-8</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1697859">1697859</a>. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0956.11021">0956.11021</a>.</cite></li>

<li><cite id="CITEREFWeil1995" class="citation cs2"><a href="Andr%C3%A9_Weil" title="André Weil">Weil, André</a> (1995), Basic number theory, Classics in Mathematics, Berlin, Heidelberg: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-58655-5</bdi></cite></li>

<li><cite id="CITEREFSerre1979" class="citation cs2"><a href="Jean-Pierre_Serre" title="Jean-Pierre Serre">Serre, Jean-Pierre</a> (1979), Local Fields, Graduate Texts in Mathematics, vol. 67 (First ed.), New York: Springer-Verlag, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-90424-7</bdi></cite></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Local_field">"Local field"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li></ul>

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