View source for Ramification theory of valued fields

A branch of [[Commutative algebra|commutative algebra]] and [[Number theory|number theory]] in which certain distinguished intermediate fields of algebraic extensions of fields equipped with a [[Valuation|valuation]] are considered. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r1100201.png" /> be a (not necessarily finite) algebraic extension of fields, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r1100202.png" /> be a valuation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r1100203.png" /> with valuation ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r1100204.png" /> and extending a valuation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r1100205.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r1100206.png" />. Assume that the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r1100207.png" /> is normal (cf. [[Extension of a field|Extension of a field]]) and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r1100208.png" /> is its [[Galois group|Galois group]]. The subgroup

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r1100209.png" /></td> </tr></table>

of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002010.png" /> is called the decomposition group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002011.png" />, and its fixed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002012.png" /> the decomposition field. The subgroup

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of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002014.png" /> is called the inertia group, and its fixed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002015.png" /> the inertia field. The subgroup

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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002017.png" /></td> </tr></table>

of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002018.png" /> is called the ramification group, and its fixed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002019.png" /> the ramification field. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002020.png" /> denotes the (unique) maximal [[Ideal|ideal]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002021.png" />, then the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002022.png" /> is equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002023.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002024.png" /> is equivalent to

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In number theory, also the higher ramification groups (cf. [[Ramified prime ideal|Ramified prime ideal]]) play a role; see [[#References|[a2]]]. If the value group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002026.png" /> is a subgroup of the real numbers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002027.png" /> is a real number, then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002028.png" />th ramification group is defined to be

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==Basic properties.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002030.png" /> denote the characteristic of the residue field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002031.png" /> if it is a positive prime number; otherwise, set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002032.png" />. For simplicity, denote the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002033.png" /> to the intermediate fields again by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002034.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002035.png" /> is a [[Pro-p group|pro-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002036.png" />-group]]; in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002037.png" /> if the characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002038.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002039.png" />. The quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002040.png" /> of the respective value groups is a [[P-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002041.png" />-group]], and the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002042.png" /> of the respective residue fields is [[Purely inseparable extension|purely inseparable]] . <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002044.png" /> are normal subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002045.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002046.png" /> is a [[Normal subgroup|normal subgroup]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002047.png" />.

The [[Galois group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002048.png" /> of the normal separable extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002049.png" /> is isomorphic to the character group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002050.png" />, which is (non-canonically) isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002051.png" /> if this group is finite. One has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002052.png" />, and the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002053.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002055.png" />-prime, i.e., no element has an order divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002056.png" />. Every finite quotient of the [[Profinite group|profinite group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002057.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002058.png" />-prime.

The Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002059.png" /> of the normal separable extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002060.png" /> is isomorphic to the Galois group of the normal extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002061.png" /> (which is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002062.png" />). Furthermore, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002063.png" /> is separable, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002064.png" />. The extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002065.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002066.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002067.png" /> is unique. The extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002068.png" /> is purely inseparable, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002069.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002070.png" />-group.

For many applications, it is more convenient to define the decomposition, inertia and ramification field to be the fixed field of the corresponding group in the maximal separable subextension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002071.png" />. Then one obtains the following additional properties: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002072.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002073.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002074.png" /> is the minimal subextension which admits a unique extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002075.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002076.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002077.png" /> is the maximal separable subextension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002078.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002079.png" /> is the maximal of all subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002080.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002081.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002082.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002083.png" />-prime.

==Absolute ramification theory.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002084.png" /> be any field with a valuation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002085.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002086.png" /> be some extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002087.png" /> to the separable-algebraic closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002088.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002089.png" />. Then the intermediate fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002090.png" /> are called the absolute decomposition field, the absolute inertia field and the absolute ramification field, respectively. Since all extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002091.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002092.png" /> are conjugate, that is, of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002093.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002094.png" />, it follows that these fields are independent of the choice of the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002095.png" />, up to isomorphism over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002096.png" />. The absolute ramification field is the Henselization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002097.png" /> inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002098.png" /> (see [[Henselization of a valued field|Henselization of a valued field]]); it coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r11002099.png" /> if and only if the extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r110020100.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r110020101.png" /> to every algebraic extension field is unique.

==Tame extensions and defectless fields.==
An extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r110020102.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r110020103.png" /> is called tamely ramified if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r110020104.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r110020105.png" />-prime and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r110020106.png" /> is separable. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r110020107.png" /> be Henselian. Then an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r110020108.png" /> is called a tame extension if it is algebraic, tamely ramified and the [[Defect|defect]] of every finite subextension is trivial, that is, equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r110020109.png" />. The absolute ramification field is the unique maximal tame extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r110020110.png" />. If it is algebraically closed, or equivalently, if all algebraic extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r110020111.png" /> are tame extensions, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r110020112.png" /> is called a tame field; see also [[Model theory of valued fields|Model theory of valued fields]]. From the fact that every finite subextension in the absolute ramification field is defectless it follows that a non-trivial defect can only appear between the absolute ramification field and the algebraic closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r110020113.png" />. Since every finite subextension of this extension has as degree a power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r110020114.png" />, the defect must be a power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r110020115.png" />. This is the content of the Ostrowski lemma. In particular, the defect is always trivial if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r110020116.png" />, that is, if the characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r110020117.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110020/r110020118.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  O. Endler,   "Valuation theory" , Springer  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.P. Serre,   "Corps locaux" , Hermann  (1962)</TD></TR></table>