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)