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 $ L \mid K $ be a (not necessarily finite) algebraic extension of fields, and let $ w $ be a valuation of $ L $ with valuation ring $ {\mathcal O} _ {w} $ and extending a valuation $ v $ of $ K $. Assume that the extension $ L \mid K $ is normal (cf. Extension of a field) and that $ G = G ( L \mid K ) $ is its Galois group. The subgroup

$$ G _ {Z} = \left \{ {\sigma \in G } : {w ( \sigma a ) = w ( a ) \textrm{ for all } a \in L } \right \} $$

of $ G $ is called the decomposition group of $ w \mid v $, and its fixed field $ Z $ the decomposition field. The subgroup

$$ G _ {T} = \left \{ {\sigma \in G } : {w ( \sigma a - a ) > 0 \textrm{ for all } a \in {\mathcal O} _ {w} } \right \} $$

of $ G _ {Z} $ is called the inertia group, and its fixed field $ T $ the inertia field. The subgroup

$$ G _ {V} = $$

$$ = \left \{ {\sigma \in G } : {w ( \sigma a - a ) > w ( a ) \textrm{ for all } a \in L, a \neq 0 } \right \} $$

of $ G _ {T} $ is called the ramification group, and its fixed field $ V $ the ramification field. If $ {\mathcal M} _ {w} $ denotes the (unique) maximal ideal of $ {\mathcal O} _ {w} $, then the condition $ w ( \sigma a - a ) > 0 $ is equivalent to $ \sigma a - a \in {\mathcal M} _ {w} $, and $ w ( \sigma a - a ) > w ( a ) $ is equivalent to

$$ { \frac{\sigma a }{a} } - 1 \in {\mathcal M} _ {w} . $$

In number theory, also the higher ramification groups (cf. Ramified prime ideal) play a role; see [a2]. If the value group $ wL $ is a subgroup of the real numbers and $ s \geq - 1 $ is a real number, then the $ s $ th ramification group is defined to be

$$ \left \{ {\sigma \in G } : {w ( \sigma a - a ) \geq s + 1 \textrm{ for all } a \in {\mathcal O} _ {w} } \right \} . $$

Basic properties.

Let $ p $ denote the characteristic of the residue field $ Lw $ if it is a positive prime number; otherwise, set $ p = 1 $. For simplicity, denote the restriction of $ w $ to the intermediate fields again by $ w $. Then $ G _ {V} $ is a pro- $ p $- group; in particular, $ L = V $ if the characteristic of $ Lw $ is $ 0 $. The quotient group $ wL/wV $ of the respective value groups is a $ p $- group, and the extension $ Lw \mid Vw $ of the respective residue fields is purely inseparable . $ G _ {V} $ and $ G _ {T} $ are normal subgroups of $ G _ {Z} $, and $ G _ {V} $ is a normal subgroup of $ G _ {T} $.

The Galois group $ G _ {T} /G _ {V} $ of the normal separable extension $ V \mid T $ is isomorphic to the character group $ { \mathop{\rm Hom} } ( wL/vK,Lw ^ \times ) $, which is (non-canonically) isomorphic to $ wV/wT $ if this group is finite. One has $ Vw = Tw $, and the group $ wV/wT $ is $ p $- prime, i.e., no element has an order divisible by $ p $. Every finite quotient of the profinite group $ G _ {T} /G _ {V} $ is $ p $- prime.

The Galois group $ G _ {Z} /G _ {T} $ of the normal separable extension $ T \mid Z $ is isomorphic to the Galois group of the normal extensions $ Lw \mid Kv $( which is $ Tw \mid Zw $). Furthermore, $ Tw \mid Zw $ is separable, and $ wT = wZ $. The extension of $ w $ from $ Z $ to $ L $ is unique. The extension $ Zw \mid Kv $ is purely inseparable, and $ wZ/vK $ is a $ p $- 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 $ L \mid K $. Then one obtains the following additional properties: $ wZ = vK $; $ Zw = Kv $; $ Z $ is the minimal subextension which admits a unique extension of $ w $ to $ L $; $ Tw \mid Kv $ is the maximal separable subextension of $ Lw \mid Kv $; and $ wV $ is the maximal of all subgroups $ \Gamma $ of $ wL $ for which $ \Gamma/vK $ is $ p $- prime.

Absolute ramification theory.

Let $ K $ be any field with a valuation $ v $, and let $ v ^ {s} $ be some extension of $ v $ to the separable-algebraic closure $ K ^ {s} $ of $ K $. Then the intermediate fields $ Z,T,V $ are called the absolute decomposition field, the absolute inertia field and the absolute ramification field, respectively. Since all extensions of $ v $ to $ K ^ {s} $ are conjugate, that is, of the form $ v \circ \sigma $ for $ \sigma \in G ( K ^ {s} \mid K ) $, it follows that these fields are independent of the choice of the extension $ v ^ {s} $, up to isomorphism over $ K $. The absolute ramification field is the Henselization of $ ( K,v ) $ inside $ ( K ^ {s} ,v ^ {s} ) $( see Henselization of a valued field); it coincides with $ K $ if and only if the extension of $ v $ from $ K $ to every algebraic extension field is unique.

Tame extensions and defectless fields.

An extension $ ( L,w ) $ of $ ( K,v ) $ is called tamely ramified if $ wL/vK $ is $ p $- prime and $ Lw \mid Kv $ is separable. Let $ ( K,v ) $ be Henselian. Then an extension of $ ( K,v ) $ is called a tame extension if it is algebraic, tamely ramified and the defect of every finite subextension is trivial, that is, equal to $ 1 $. The absolute ramification field is the unique maximal tame extension of $ ( K,v ) $. If it is algebraically closed, or equivalently, if all algebraic extensions of $ ( K,v ) $ are tame extensions, then $ ( K,v ) $ 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 $ K $. Since every finite subextension of this extension has as degree a power of $ p $, the defect must be a power of $ p $. This is the content of the Ostrowski lemma. In particular, the defect is always trivial if $ p = 1 $, that is, if the characteristic of $ Kv $ is $ 0 $.

References

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