In number theory, more specifically in local class field theory, the ramification groups are a filtration of the Galois group of a local field extension, which gives detailed information on the ramification phenomena of the extension.

Ramification theory of valuations

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In mathematics, the ramification theory of valuations studies the set of extensions of a valuation v of a field K to an extension L of K. It is a generalization of the ramification theory of Dedekind domains.[1][2]

The structure of the set of extensions is known better when L/K is Galois.

Decomposition group and inertia group

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Let (Kv) be a valued field and let L be a finite Galois extension of K. Let Sv be the set of equivalence classes of extensions of v to L and let G be the Galois group of L over K. Then G acts on Sv by σ[w] = [w ∘ σ] (i.e. w is a representative of the equivalence class [w] ∈ Sv and [w] is sent to the equivalence class of the composition of w with the automorphism σ : LL; this is independent of the choice of w in [w]). In fact, this action is transitive.

Given a fixed extension w of v to L, the decomposition group of w is the stabilizer subgroup Gw of [w], i.e. it is the subgroup of G consisting of all elements that fix the equivalence class [w] ∈ Sv.

Let mw denote the maximal ideal of w inside the valuation ring Rw of w. The inertia group of w is the subgroup Iw of Gw consisting of elements σ such that σx ≡ x (mod mw) for all x in Rw. In other words, Iw consists of the elements of the decomposition group that act trivially on the residue field of w. It is a normal subgroup of Gw.

The reduced ramification index e(w/v) is independent of w and is denoted e(v). Similarly, the relative degree f(w/v) is also independent of w and is denoted f(v).

Ramification groups in lower numbering

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Ramification groups are a refinement of the Galois group   of a finite   Galois extension of local fields. We shall write   for the valuation, the ring of integers and its maximal ideal for  . As a consequence of Hensel's lemma, one can write   for some   where   is the ring of integers of  .[3] (This is stronger than the primitive element theorem.) Then, for each integer  , we define   to be the set of all   that satisfies the following equivalent conditions.

  • (i)   operates trivially on  
  • (ii)   for all  
  • (iii)  

The group   is called  -th ramification group. They form a decreasing filtration,

 

In fact, the   are normal by (i) and trivial for sufficiently large   by (iii). For the lowest indices, it is customary to call   the inertia subgroup of   because of its relation to splitting of prime ideals, while   the wild inertia subgroup of  . The quotient   is called the tame quotient.

The Galois group   and its subgroups   are studied by employing the above filtration or, more specifically, the corresponding quotients. In particular,

  •   where   are the (finite) residue fields of  .[4]
  •   is unramified.
  •   is tamely ramified (i.e., the ramification index is prime to the residue characteristic.)

The study of ramification groups reduces to the totally ramified case since one has   for  .

One also defines the function  . (ii) in the above shows   is independent of choice of   and, moreover, the study of the filtration   is essentially equivalent to that of  .[5]   satisfies the following: for  ,

  •  
  •  
  •  

Fix a uniformizer   of  . Then   induces the injection   where  . (The map actually does not depend on the choice of the uniformizer.[6]) It follows from this[7]

  •   is cyclic of order prime to  
  •   is a product of cyclic groups of order  .

In particular,   is a p-group and   is solvable.

The ramification groups can be used to compute the different   of the extension   and that of subextensions:[8]

 

If   is a normal subgroup of  , then, for  ,  .[9]

Combining this with the above one obtains: for a subextension   corresponding to  ,

 

If  , then  .[10] In the terminology of Lazard, this can be understood to mean the Lie algebra   is abelian.

Example: the cyclotomic extension

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The ramification groups for a cyclotomic extension  , where   is a  -th primitive root of unity, can be described explicitly:[11]

 

where e is chosen such that  .

Example: a quartic extension

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Let K be the extension of Q2 generated by  . The conjugates of   are  ,  ,  .

A little computation shows that the quotient of any two of these is a unit. Hence they all generate the same ideal; call it π.   generates π2; (2)=π4.

Now  , which is in π5.

and   which is in π3.

Various methods show that the Galois group of K is  , cyclic of order 4. Also:

 

and  

  so that the different  

  satisfies X4 − 4X2 + 2, which has discriminant 2048 = 211.

Ramification groups in upper numbering

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If   is a real number  , let   denote   where i the least integer  . In other words,   Define   by[12]

 

where, by convention,   is equal to   if   and is equal to   for  .[13] Then   for  . It is immediate that   is continuous and strictly increasing, and thus has the continuous inverse function   defined on  . Define  .   is then called the v-th ramification group in upper numbering. In other words,  . Note  . The upper numbering is defined so as to be compatible with passage to quotients:[14] if   is normal in  , then

  for all  

(whereas lower numbering is compatible with passage to subgroups.)

Herbrand's theorem

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Herbrand's theorem states that the ramification groups in the lower numbering satisfy   (for   where   is the subextension corresponding to  ), and that the ramification groups in the upper numbering satisfy  .[15][16] This allows one to define ramification groups in the upper numbering for infinite Galois extensions (such as the absolute Galois group of a local field) from the inverse system of ramification groups for finite subextensions.

The upper numbering for an abelian extension is important because of the Hasse–Arf theorem. It states that if   is abelian, then the jumps in the filtration   are integers; i.e.,   whenever   is not an integer.[17]

The upper numbering is compatible with the filtration of the norm residue group by the unit groups under the Artin isomorphism. The image of   under the isomorphism

 

is just[18]

 

See also

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Notes

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  1. ^ Fröhlich, A.; Taylor, M.J. (1991). Algebraic number theory. Cambridge studies in advanced mathematics. Vol. 27. Cambridge University Press. ISBN 0-521-36664-X. Zbl 0744.11001.
  2. ^ Zariski, Oscar; Samuel, Pierre (1976) [1960]. Commutative algebra, Volume II. Graduate Texts in Mathematics. Vol. 29. New York, Heidelberg: Springer-Verlag. Chapter VI. ISBN 978-0-387-90171-8. Zbl 0322.13001.
  3. ^ Neukirch (1999) p.178
  4. ^ since   is canonically isomorphic to the decomposition group.
  5. ^ Serre (1979) p.62
  6. ^ Conrad
  7. ^ Use   and  
  8. ^ Serre (1979) 4.1 Prop.4, p.64
  9. ^ Serre (1979) 4.1. Prop.3, p.63
  10. ^ Serre (1979) 4.2. Proposition 10.
  11. ^ Serre, Corps locaux. Ch. IV, §4, Proposition 18
  12. ^ Serre (1967) p.156
  13. ^ Neukirch (1999) p.179
  14. ^ Serre (1967) p.155
  15. ^ Neukirch (1999) p.180
  16. ^ Serre (1979) p.75
  17. ^ Neukirch (1999) p.355
  18. ^ Snaith (1994) pp.30-31

References

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