# Defect(2)

ramification deficiency

An invariant of finite extensions $L \mid K$ of fields equipped with a valuation (cf. also Extension of a field). If a valuation $w$ on $L$ is the unique extension of a valuation $v$ on $K$, then the defect (or ramification deficiency) $d = d ( w \mid v )$ is defined by the formula $[ L:K ] = d \cdot e \cdot f$, where $[ L:K ]$ is the degree of $L \mid K$( i.e., the dimension of $L$ as a $K$- vector space), $e = e ( w \mid v ) = ( wL:vK )$ is the ramification index and $f = f ( w \mid v ) = [ Lw:Kv ]$ is the inertia degree. Here, $wL,vK$ denote the respective value groups and $Lw,Kv$ the respective residue fields. If $v$ admits several extensions to $L$, the defect $d = d ( w \mid v )$ can be defined by $[ L:K ] = d \cdot e \cdot f \cdot g$, where $g$ is the number of distinct extensions, provided that $L \mid K$ is normal (since in that case the ramification index and the inertia degree are the same for all extensions; cf. also Normal extension).

In the above cases, $e$, $f$, $g$ are divisors of $[ L:K ]$. The defect $d$ is either equal to $1$ or is a power of the characteristic $p$ of $Kv$ if $p > 0$; otherwise, it is always equal to $1$( this is the Ostrowski lemma, cf. Ramification theory of valued fields).

## Henselian defect.

To avoid considering several valuations and to have a defect available for arbitrary finite extensions, one can pass to a Henselization $( L ^ {H} ,w ^ {H} )$ of $( L,w )$ and a Henselization $( K ^ {H} ,v ^ {H} )$ of $( K,v )$ inside $( L ^ {H} ,w ^ {H} )$( cf. Henselization of a valued field). The Henselian defect $\delta ( w \mid v )$ is then defined to be the defect of $w ^ {H} \mid v ^ {H}$( by the definition of the Henselization, $w ^ {H}$ is the unique extension of $v ^ {H}$). In the above cases, $\delta ( w \mid v ) = d ( w \mid v )$.

## Defectless fields.

A field $K$ with a valuation $v$ is called a defectless field if $d ( w \mid v ) = 1$ for every finite normal extension. This holds if and only if the Henselian defect is equal to $1$ for every finite extension. It follows that $( K,v )$ is a defectless field if and only some Henselization of $( K,v )$ is (or equivalently, all Henselizations are).

It follows from the Ostrowski lemma that all valued fields with residue field of characteristic $0$ are defectless fields. Also, valued fields of characteristic $0$ with value group isomorphic to $\mathbf Z$ are defectless. Combining both facts, it is shown that finitely ramified fields, and hence also fields with $p$- valuations (see $p$- adically closed field), are defectless.

If a valued field does not admit any non-trivial immediate extension (cf. also Valuation), then it is called a maximal valued field. Fields of formal Laurent series with their canonical valuations are maximal. Every maximal valued field is defectless.

## Fundamental inequality.

If $w _ {1} \dots w _ {m}$ are all extensions of $v$ from $K$ to $L$, then one has the fundamental inequality

$$[ L:K ] \geq \sum _ {i = 1 } ^ { m } e ( w _ {i} \mid v ) \cdot f ( w _ {i} \mid w ) .$$

This is an equality for every finite $L \mid K$ if and only if $( K,v )$ is defectless. Also, in general it can be written as an equality. For this, choose Henselizations $( L ^ {H _ {i} } ,w ^ {H _ {i} } )$ of $( L,w _ {i} )$ and $( K ^ {H _ {i} } ,v ^ {H _ {i} } )$ of $( K,v )$ inside $( L ^ {H _ {i} } ,w ^ {H _ {i} } )$. It is known that

$$[ L:K ] = \sum _ {i = 1 } ^ { m } [ L ^ {H _ {i} } :K ^ {H _ {i} } ] .$$

Further,

$$[ L ^ {H _ {i} } :K ^ {H _ {i} } ] =$$

$$= d ( w ^ {H _ {i} } \mid v ^ {H _ {i} } ) \cdot e ( w ^ {H _ {i} } \mid v ^ {H _ {i} } ) \cdot f ( w ^ {H _ {i} } \mid v ^ {H _ {i} } ) .$$

Since Henselizations are immediate extensions, $e ( w ^ {H _ {i} } \mid v ^ {H _ {i} } ) = e ( w \mid v )$ and $f ( w ^ {H _ {i} } \mid v ^ {H _ {i} } ) = f ( w \mid v )$. By definition, $d ( w ^ {H _ {i} } \mid v ^ {H _ {i} } ) = \delta ( w _ {i} \mid v )$. Hence,

$$[ L:K ] = \sum _ {i = 1 } ^ { m } \delta ( w _ {i} \mid v ) \cdot e ( w _ {i} \mid v ) \cdot f ( w _ {i} \mid w ) .$$

Several other notions of defects were introduced. For a detailed theory of the defect, see [a1]. See also Valued function field.

#### References

 [a1] F.-V. Kuhlmann, "Valuation theory of fields, abelian groups and modules" , Algebra, Logic and Applications , Gordon&Breach (to appear)
How to Cite This Entry:
Defect(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Defect(2)&oldid=46600
This article was adapted from an original article by F.-V. Kuhlmann (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article