Ramification theory of valued fields

A branch of commutative algebra and number theory in which certain distinguished intermediate fields of algebraic extensions of fields equipped with a valuation are considered. Let be a (not necessarily finite) algebraic extension of fields, and let be a valuation of with valuation ring and extending a valuation of . Assume that the extension is normal (cf. Extension of a field) and that is its Galois group. The subgroup

of is called the decomposition group of , and its fixed field the decomposition field. The subgroup

of is called the inertia group, and its fixed field the inertia field. The subgroup

of is called the ramification group, and its fixed field the ramification field. If denotes the (unique) maximal ideal of , then the condition is equivalent to , and is equivalent to

In number theory, also the higher ramification groups (cf. Ramified prime ideal) play a role; see [a2]. If the value group is a subgroup of the real numbers and is a real number, then the th ramification group is defined to be

Basic properties.

Let denote the characteristic of the residue field if it is a positive prime number; otherwise, set . For simplicity, denote the restriction of to the intermediate fields again by . Then is a pro--group; in particular, if the characteristic of is . The quotient group of the respective value groups is a -group, and the extension of the respective residue fields is purely inseparable (cf. Separable extension). and are normal subgroups of , and is a normal subgroup of .

The Galois group of the normal separable extension is isomorphic to the character group , which is (non-canonically) isomorphic to if this group is finite. One has , and the group is -prime, i.e., no element has an order divisible by . Every finite quotient of the profinite group is -prime.

The Galois group of the normal separable extension is isomorphic to the Galois group of the normal extensions (which is ). Furthermore, is separable, and . The extension of from to is unique. The extension is purely inseparable, and is a -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 . Then one obtains the following additional properties: ; ; is the minimal subextension which admits a unique extension of to ; is the maximal separable subextension of ; and is the maximal of all subgroups of for which is -prime.

Absolute ramification theory.

Let be any field with a valuation , and let be some extension of to the separable-algebraic closure of . Then the intermediate fields are called the absolute decomposition field, the absolute inertia field and the absolute ramification field, respectively. Since all extensions of to are conjugate, that is, of the form for , it follows that these fields are independent of the choice of the extension , up to isomorphism over . The absolute ramification field is the Henselization of inside (see Henselization of a valued field); it coincides with if and only if the extension of from to every algebraic extension field is unique.

Tame extensions and defectless fields.

An extension of is called tamely ramified if is -prime and is separable. Let be Henselian. Then an extension of is called a tame extension if it is algebraic, tamely ramified and the defect of every finite subextension is trivial, that is, equal to . The absolute ramification field is the unique maximal tame extension of . If it is algebraically closed, or equivalently, if all algebraic extensions of are tame extensions, then is called a tame field; see also 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 . Since every finite subextension of this extension has as degree a power of , the defect must be a power of . This is the content of the Ostrowski lemma. In particular, the defect is always trivial if , that is, if the characteristic of is .

References

[a1]	O. Endler, "Valuation theory" , Springer (1972)
[a2]	J.P. Serre, "Corps locaux" , Hermann (1962)