# Ramification group

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Filtration of the Galois group of a local field extension

In [number theory](/source/Number_theory), more specifically in [local class field theory](/source/Local_class_field_theory), the **ramification groups** are a [filtration](/source/Filtration_(mathematics)) of the [Galois group](/source/Galois_group) of a [local field](/source/Local_field) extension, which gives detailed information on the [ramification](/source/Ramification_(mathematics)) phenomena of the extension.

## Ramification theory of valuations

In [mathematics](/source/Mathematics), the **ramification theory of valuations** studies the set of [extensions](/source/Extension_of_a_valuation) of a [valuation](/source/Valuation_(algebra)) *v* of a [field](/source/Field_(mathematics)) *K* to an [extension](/source/Field_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](/source/Galois_extension).

### Decomposition group and inertia group

Let (*K*, *v*) be a [valued field](/source/Valued_field) and let *L* be a [finite](/source/Finite_extension) [Galois extension](/source/Galois_extension) of *K*. Let *Sv* be the set of [equivalence](/source/Equivalence_of_valuations) [classes](/source/Equivalence_class) of extensions of *v* to *L* and let *G* be the [Galois group](/source/Galois_group) of *L* over *K*. Then *G* acts on *Sv* by σ[*w*] = [*w* ∘ σ] (i.e. *w* is a [representative](/source/Representative_(mathematics)) of the equivalence class [*w*] ∈ *Sv* and [*w*] is sent to the equivalence class of the [composition](/source/Function_composition) of *w* with the [automorphism](/source/Automorphism) σ : *L* → *L*; this is independent of the choice of *w* in [*w*]). In fact, this action is [transitive](/source/Transitive_action).

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

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

The [reduced ramification index](/source/Reduced_ramification_index_of_an_extension_of_valuations) *e*(*w*/*v*) is independent of *w* and is denoted *e*(*v*). Similarly, the [relative degree](/source/Relative_degree_of_an_extension_of_valuations) *f*(*w*/*v*) is also independent of *w* and is denoted *f*(*v*).

## Ramification groups in lower numbering

Ramification groups are a refinement of the Galois group G {\displaystyle G} of a finite L / K {\displaystyle L/K} [Galois extension](/source/Galois_extension) of [local fields](/source/Local_field). We shall write w , O L , p {\displaystyle w,{\mathcal {O}}_{L},{\mathfrak {p}}} for the valuation, the ring of integers and its maximal ideal for L {\displaystyle L} . As a consequence of [Hensel's lemma](/source/Hensel's_lemma), one can write O L = O K [ α ] {\displaystyle {\mathcal {O}}_{L}={\mathcal {O}}_{K}[\alpha ]} for some α ∈ L {\displaystyle \alpha \in L} where O K {\displaystyle {\mathcal {O}}_{K}} is the ring of integers of K {\displaystyle K} .[3] (This is stronger than the [primitive element theorem](/source/Primitive_element_theorem).) Then, for each integer i ≥ − 1 {\displaystyle i\geq -1} , we define G i {\displaystyle G_{i}} to be the set of all s ∈ G {\displaystyle s\in G} that satisfies the following equivalent conditions.

- (i) s {\displaystyle s} operates trivially on O L / p i + 1 . {\displaystyle {\mathcal {O}}_{L}/{\mathfrak {p}}^{i+1}.}

- (ii) w ( s ( x ) − x ) ≥ i + 1 {\displaystyle w(s(x)-x)\geq i+1} for all x ∈ O L {\displaystyle x\in {\mathcal {O}}_{L}}

- (iii) w ( s ( α ) − α ) ≥ i + 1. {\displaystyle w(s(\alpha )-\alpha )\geq i+1.}

The group G i {\displaystyle G_{i}} is called *i {\displaystyle i} -th ramification group*. They form a decreasing [filtration](/source/Filtration_(mathematics)),

- G − 1 = G ⊃ G 0 ⊃ G 1 ⊃ … { ∗ } . {\displaystyle G_{-1}=G\supset G_{0}\supset G_{1}\supset \dots \{*\}.}

In fact, the G i {\displaystyle G_{i}} are normal by (i) and [trivial](/source/Trivial_group) for sufficiently large i {\displaystyle i} by (iii). For the lowest indices, it is customary to call G 0 {\displaystyle G_{0}} the [inertia subgroup](/source/Inertia_subgroup) of G {\displaystyle G} because of its relation to [splitting of prime ideals](/source/Splitting_of_prime_ideals_in_Galois_extensions), while G 1 {\displaystyle G_{1}} the [wild inertia subgroup](https://en.wikipedia.org/w/index.php?title=Wild_inertia_subgroup&action=edit&redlink=1) of G {\displaystyle G} . The quotient G 0 / G 1 {\displaystyle G_{0}/G_{1}} is called the tame quotient.

The Galois group G {\displaystyle G} and its subgroups G i {\displaystyle G_{i}} are studied by employing the above filtration or, more specifically, the corresponding quotients. In particular,

- G / G 0 = Gal ⁡ ( l / k ) , {\displaystyle G/G_{0}=\operatorname {Gal} (l/k),} where l , k {\displaystyle l,k} are the (finite) residue fields of L , K {\displaystyle L,K} .[4]

- G 0 = 1 ⇔ L / K {\displaystyle G_{0}=1\Leftrightarrow L/K} is [unramified](/source/Unramified_extension).

- G 1 = 1 ⇔ L / K {\displaystyle G_{1}=1\Leftrightarrow L/K} is [tamely ramified](/source/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 G i = ( G 0 ) i {\displaystyle G_{i}=(G_{0})_{i}} for i ≥ 0 {\displaystyle i\geq 0} .

One also defines the function i G ( s ) = w ( s ( α ) − α ) , s ∈ G {\displaystyle i_{G}(s)=w(s(\alpha )-\alpha ),s\in G} . (ii) in the above shows i G {\displaystyle i_{G}} is independent of choice of α {\displaystyle \alpha } and, moreover, the study of the filtration G i {\displaystyle G_{i}} is essentially equivalent to that of i G {\displaystyle i_{G}} .[5] i G {\displaystyle i_{G}} satisfies the following: for s , t ∈ G {\displaystyle s,t\in G} ,

- i G ( s ) ≥ i + 1 ⇔ s ∈ G i . {\displaystyle i_{G}(s)\geq i+1\Leftrightarrow s\in G_{i}.}

- i G ( t s t − 1 ) = i G ( s ) . {\displaystyle i_{G}(tst^{-1})=i_{G}(s).}

- i G ( s t ) ≥ min { i G ( s ) , i G ( t ) } . {\displaystyle i_{G}(st)\geq \min\{i_{G}(s),i_{G}(t)\}.}

Fix a uniformizer π {\displaystyle \pi } of L {\displaystyle L} . Then s ↦ s ( π ) / π {\displaystyle s\mapsto s(\pi )/\pi } induces the injection G i / G i + 1 → U L , i / U L , i + 1 , i ≥ 0 {\displaystyle G_{i}/G_{i+1}\to U_{L,i}/U_{L,i+1},i\geq 0} where U L , 0 = O L × , U L , i = 1 + p i {\displaystyle U_{L,0}={\mathcal {O}}_{L}^{\times },U_{L,i}=1+{\mathfrak {p}}^{i}} . (The map actually does not depend on the choice of the uniformizer.[6]) It follows from this[7]

- G 0 / G 1 {\displaystyle G_{0}/G_{1}} is cyclic of order prime to p {\displaystyle p}

- G i / G i + 1 {\displaystyle G_{i}/G_{i+1}} is a product of cyclic groups of order p {\displaystyle p} .

In particular, G 1 {\displaystyle G_{1}} is a [*p*-group](/source/P-group) and G 0 {\displaystyle G_{0}} is [solvable](/source/Solvable_group).

The ramification groups can be used to compute the [different](/source/Different_ideal) D L / K {\displaystyle {\mathfrak {D}}_{L/K}} of the extension L / K {\displaystyle L/K} and that of subextensions:[8]

- w ( D L / K ) = ∑ s ≠ 1 i G ( s ) = ∑ i = 0 ∞ ( | G i | − 1 ) . {\displaystyle w({\mathfrak {D}}_{L/K})=\sum _{s\neq 1}i_{G}(s)=\sum _{i=0}^{\infty }(|G_{i}|-1).}

If H {\displaystyle H} is a normal subgroup of G {\displaystyle G} , then, for σ ∈ G {\displaystyle \sigma \in G} , i G / H ( σ ) = 1 e L / K ∑ s ↦ σ i G ( s ) {\displaystyle i_{G/H}(\sigma )={1 \over e_{L/K}}\sum _{s\mapsto \sigma }i_{G}(s)} .[9]

Combining this with the above one obtains: for a subextension F / K {\displaystyle F/K} corresponding to H {\displaystyle H} ,

- v F ( D F / K ) = 1 e L / F ∑ s ∉ H i G ( s ) . {\displaystyle v_{F}({\mathfrak {D}}_{F/K})={1 \over e_{L/F}}\sum _{s\not \in H}i_{G}(s).}

If s ∈ G i , t ∈ G j , i , j ≥ 1 {\displaystyle s\in G_{i},t\in G_{j},i,j\geq 1} , then s t s − 1 t − 1 ∈ G i + j + 1 {\displaystyle sts^{-1}t^{-1}\in G_{i+j+1}} .[10] In the terminology of [Lazard](/source/Michel_Lazard), this can be understood to mean the [Lie algebra](/source/Lie_algebra) gr ⁡ ( G 1 ) = ∑ i ≥ 1 G i / G i + 1 {\displaystyle \operatorname {gr} (G_{1})=\sum _{i\geq 1}G_{i}/G_{i+1}} is abelian.

### Example: the cyclotomic extension

The ramification groups for a [cyclotomic extension](/source/Cyclotomic_extension) K n := Q p ( ζ ) / Q p {\displaystyle K_{n}:=\mathbf {Q} _{p}(\zeta )/\mathbf {Q} _{p}} , where ζ {\displaystyle \zeta } is a p n {\displaystyle p^{n}} -th primitive [root of unity](/source/Root_of_unity), can be described explicitly:[11]

- G s = Gal ⁡ ( K n / K e ) , {\displaystyle G_{s}=\operatorname {Gal} (K_{n}/K_{e}),}

where *e* is chosen such that p e − 1 ≤ s < p e {\displaystyle p^{e-1}\leq s<p^{e}} .

### Example: a quartic extension

Let *K* be the extension of **Q**2 generated by x 1 = 2 + 2 {\displaystyle x_{1}={\sqrt {2+{\sqrt {2}}}}} . The conjugates of x 1 {\displaystyle x_{1}} are x 2 = 2 − 2 {\displaystyle x_{2}={\sqrt {2-{\sqrt {2}}}}} , x 3 = − x 1 {\displaystyle x_{3}=-x_{1}} , x 4 = − x 2 {\displaystyle x_{4}=-x_{2}} .

A little computation shows that the quotient of any two of these is a [unit](/source/Unit_(ring_theory)). Hence they all generate the same ideal; call it π. 2 {\displaystyle {\sqrt {2}}} generates π2; (2)=π4.

Now x 1 − x 3 = 2 x 1 {\displaystyle x_{1}-x_{3}=2x_{1}} , which is in π5.

and x 1 − x 2 = 4 − 2 2 , {\displaystyle x_{1}-x_{2}={\sqrt {4-2{\sqrt {2}}}},} which is in π3.

Various methods show that the Galois group of *K* is C 4 {\displaystyle C_{4}} , cyclic of order 4. Also:

- G 0 = G 1 = G 2 = C 4 . {\displaystyle G_{0}=G_{1}=G_{2}=C_{4}.}

and G 3 = G 4 = ( 13 ) ( 24 ) . {\displaystyle G_{3}=G_{4}=(13)(24).}

w ( D K / Q 2 ) = 3 + 3 + 3 + 1 + 1 = 11 , {\displaystyle w({\mathfrak {D}}_{K/Q_{2}})=3+3+3+1+1=11,} so that the different D K / Q 2 = π 11 {\displaystyle {\mathfrak {D}}_{K/Q_{2}}=\pi ^{11}}

x 1 {\displaystyle x_{1}} satisfies *X*4 − 4*X*2 + 2, which has discriminant 2048 = 211.

## Ramification groups in upper numbering

If u {\displaystyle u} is a real number ≥ − 1 {\displaystyle \geq -1} , let G u {\displaystyle G_{u}} denote G i {\displaystyle G_{i}} where *i* the least integer ≥ u {\displaystyle \geq u} . In other words, s ∈ G u ⇔ i G ( s ) ≥ u + 1. {\displaystyle s\in G_{u}\Leftrightarrow i_{G}(s)\geq u+1.} Define ϕ {\displaystyle \phi } by[12]

- ϕ ( u ) = ∫ 0 u d t ( G 0 : G t ) {\displaystyle \phi (u)=\int _{0}^{u}{dt \over (G_{0}:G_{t})}}

where, by convention, ( G 0 : G t ) {\displaystyle (G_{0}:G_{t})} is equal to ( G − 1 : G 0 ) − 1 {\displaystyle (G_{-1}:G_{0})^{-1}} if t = − 1 {\displaystyle t=-1} and is equal to 1 {\displaystyle 1} for − 1 < t ≤ 0 {\displaystyle -1<t\leq 0} .[13] Then ϕ ( u ) = u {\displaystyle \phi (u)=u} for − 1 ≤ u ≤ 0 {\displaystyle -1\leq u\leq 0} . It is immediate that ϕ {\displaystyle \phi } is continuous and strictly increasing, and thus has the continuous inverse function ψ {\displaystyle \psi } defined on [ − 1 , ∞ ) {\displaystyle [-1,\infty )} . Define G v = G ψ ( v ) {\displaystyle G^{v}=G_{\psi (v)}} . G v {\displaystyle G^{v}} is then called the ***v*-th ramification group** in upper numbering. In other words, G ϕ ( u ) = G u {\displaystyle G^{\phi (u)}=G_{u}} . Note G − 1 = G , G 0 = G 0 {\displaystyle G^{-1}=G,G^{0}=G_{0}} . The upper numbering is defined so as to be compatible with passage to quotients:[14] if H {\displaystyle H} is normal in G {\displaystyle G} , then

- ( G / H ) v = G v H / H {\displaystyle (G/H)^{v}=G^{v}H/H} for all v {\displaystyle v}

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

### Herbrand's theorem

**Herbrand's theorem** states that the ramification groups in the lower numbering satisfy G u H / H = ( G / H ) v {\displaystyle G_{u}H/H=(G/H)_{v}} (for v = ϕ L / F ( u ) {\displaystyle v=\phi _{L/F}(u)} where L / F {\displaystyle L/F} is the subextension corresponding to H {\displaystyle H} ), and that the ramification groups in the upper numbering satisfy G u H / H = ( G / H ) u {\displaystyle G^{u}H/H=(G/H)^{u}} .[15][16] This allows one to define ramification groups in the upper numbering for infinite Galois extensions (such as the [absolute Galois group](/source/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](/source/Hasse%E2%80%93Arf_theorem). It states that if G {\displaystyle G} is abelian, then the jumps in the filtration G v {\displaystyle G^{v}} are integers; i.e., G i = G i + 1 {\displaystyle G_{i}=G_{i+1}} whenever ϕ ( i ) {\displaystyle \phi (i)} 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](/source/Artin_isomorphism). The image of G n ( L / K ) {\displaystyle G^{n}(L/K)} under the isomorphism

- G ( L / K ) a b ↔ K ∗ / N L / K ( L ∗ ) {\displaystyle G(L/K)^{\mathrm {ab} }\leftrightarrow K^{*}/N_{L/K}(L^{*})}

is just[18]

- U K n / ( U K n ∩ N L / K ( L ∗ ) ) . {\displaystyle U_{K}^{n}/(U_{K}^{n}\cap N_{L/K}(L^{*}))\ .}

## See also

- [Finite extensions of local fields](/source/Finite_extensions_of_local_fields)

## Notes

1. **[^](#cite_ref-1)** [Fröhlich, A.](/source/Albrecht_Fr%C3%B6hlich); [Taylor, M.J.](/source/Martin_J._Taylor) (1991). *Algebraic number theory*. Cambridge studies in advanced mathematics. Vol. 27. [Cambridge University Press](/source/Cambridge_University_Press). [ISBN](/source/ISBN_(identifier)) [0-521-36664-X](https://en.wikipedia.org/wiki/Special:BookSources/0-521-36664-X). [Zbl](/source/Zbl_(identifier)) [0744.11001](https://zbmath.org/?format=complete&q=an:0744.11001).

1. **[^](#cite_ref-2)** [Zariski, Oscar](/source/Oscar_Zariski); [Samuel, Pierre](/source/Pierre_Samuel) (1976) [1960]. *Commutative algebra, Volume II*. [Graduate Texts in Mathematics](/source/Graduate_Texts_in_Mathematics). Vol. 29. New York, Heidelberg: Springer-Verlag. Chapter VI. [ISBN](/source/ISBN_(identifier)) [978-0-387-90171-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-90171-8). [Zbl](/source/Zbl_(identifier)) [0322.13001](https://zbmath.org/?format=complete&q=an:0322.13001).

1. **[^](#cite_ref-N178_3-0)** Neukirch (1999) p.178

1. **[^](#cite_ref-4)** since G / G 0 {\displaystyle G/G_{0}} is canonically isomorphic to the decomposition group.

1. **[^](#cite_ref-S7962_5-0)** Serre (1979) p.62

1. **[^](#cite_ref-6)** Conrad

1. **[^](#cite_ref-7)** Use U L , 0 / U L , 1 ≃ l × {\displaystyle U_{L,0}/U_{L,1}\simeq l^{\times }} and U L , i / U L , i + 1 ≈ l + {\displaystyle U_{L,i}/U_{L,i+1}\approx l^{+}}

1. **[^](#cite_ref-S64_8-0)** Serre (1979) 4.1 Prop.4, p.64

1. **[^](#cite_ref-S63_9-0)** Serre (1979) 4.1. Prop.3, p.63

1. **[^](#cite_ref-10)** Serre (1979) 4.2. Proposition 10.

1. **[^](#cite_ref-11)** Serre, *Corps locaux*. Ch. IV, §4, Proposition 18

1. **[^](#cite_ref-S67156_12-0)** Serre (1967) p.156

1. **[^](#cite_ref-N179_13-0)** Neukirch (1999) p.179

1. **[^](#cite_ref-S67155_14-0)** Serre (1967) p.155

1. **[^](#cite_ref-N180_15-0)** Neukirch (1999) p.180

1. **[^](#cite_ref-S75_16-0)** Serre (1979) p.75

1. **[^](#cite_ref-N355_17-0)** Neukirch (1999) p.355

1. **[^](#cite_ref-Sn3031_18-0)** Snaith (1994) pp.30-31

## References

- B. Conrad, [Math 248A. Higher ramification groups](http://math.stanford.edu/~conrad/248APage/handouts/ramgroup.pdf)

- [Fröhlich, A.](/source/Albrecht_Fr%C3%B6hlich); [Taylor, M.J.](/source/Martin_J._Taylor) (1991). *Algebraic number theory*. Cambridge studies in advanced mathematics. Vol. 27. [Cambridge University Press](/source/Cambridge_University_Press). [ISBN](/source/ISBN_(identifier)) [0-521-36664-X](https://en.wikipedia.org/wiki/Special:BookSources/0-521-36664-X). [Zbl](/source/Zbl_(identifier)) [0744.11001](https://zbmath.org/?format=complete&q=an:0744.11001).

- [Neukirch, Jürgen](/source/J%C3%BCrgen_Neukirch) (1999). *Algebraische Zahlentheorie*. *Grundlehren der mathematischen Wissenschaften*. Vol. 322. Berlin: [Springer-Verlag](/source/Springer_Science%2BBusiness_Media). [ISBN](/source/ISBN_(identifier)) [978-3-540-65399-8](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-65399-8). [MR](/source/MR_(identifier)) [1697859](https://mathscinet.ams.org/mathscinet-getitem?mr=1697859). [Zbl](/source/Zbl_(identifier)) [0956.11021](https://zbmath.org/?format=complete&q=an:0956.11021).

- [Serre, Jean-Pierre](/source/Jean-Pierre_Serre) (1967). "VI. Local class field theory". In [Cassels, J.W.S.](/source/J._W._S._Cassels); [Fröhlich, A.](/source/Albrecht_Fr%C3%B6hlich) (eds.). *Algebraic number theory. Proceedings of an instructional conference organized by the London Mathematical Society (a NATO Advanced Study Institute) with the support of the International Mathematical Union*. London: Academic Press. pp. 128–161. [Zbl](/source/Zbl_(identifier)) [0153.07403](https://zbmath.org/?format=complete&q=an:0153.07403).

- [Serre, Jean-Pierre](/source/Jean-Pierre_Serre) (1979). *[Local Fields](/source/Local_Fields)*. Graduate Texts in Mathematics. Vol. 67. Translated by [Greenberg, Marvin Jay](/source/Marvin_Greenberg). Berlin, New York: [Springer-Verlag](/source/Springer-Verlag). [ISBN](/source/ISBN_(identifier)) [0-387-90424-7](https://en.wikipedia.org/wiki/Special:BookSources/0-387-90424-7). [MR](/source/MR_(identifier)) [0554237](https://mathscinet.ams.org/mathscinet-getitem?mr=0554237). [Zbl](/source/Zbl_(identifier)) [0423.12016](https://zbmath.org/?format=complete&q=an:0423.12016).

- Snaith, Victor P. (1994). *Galois module structure*. Fields Institute monographs. Providence, RI: [American Mathematical Society](/source/American_Mathematical_Society). [ISBN](/source/ISBN_(identifier)) [0-8218-0264-X](https://en.wikipedia.org/wiki/Special:BookSources/0-8218-0264-X). [Zbl](/source/Zbl_(identifier)) [0830.11042](https://zbmath.org/?format=complete&q=an:0830.11042).

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