Defect(2)

ramification deficiency

An invariant of finite extensions of fields equipped with a valuation (cf. also Extension of a field). If a valuation on is the unique extension of a valuation on , then the defect (or ramification deficiency) is defined by the formula , where is the degree of (i.e., the dimension of as a -vector space), is the ramification index and is the inertia degree. Here, denote the respective value groups and the respective residue fields. If admits several extensions to , the defect can be defined by , where is the number of distinct extensions, provided that 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, , , are divisors of . The defect is either equal to or is a power of the characteristic of if ; otherwise, it is always equal to (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 of and a Henselization of inside (cf. Henselization of a valued field). The Henselian defect is then defined to be the defect of (by the definition of the Henselization, is the unique extension of ). In the above cases, .

Defectless fields.

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

It follows from the Ostrowski lemma that all valued fields with residue field of characteristic are defectless fields. Also, valued fields of characteristic with value group isomorphic to are defectless. Combining both facts, it is shown that finitely ramified fields, and hence also fields with -valuations (see -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 are all extensions of from to , then one has the fundamental inequality

This is an equality for every finite if and only if is defectless. Also, in general it can be written as an equality. For this, choose Henselizations of and of inside . It is known that

Further,

Since Henselizations are immediate extensions, and . By definition, . Hence,

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)