# Ramification group

Ramification groups form a descending filtration of the [Galois group](https://www.edgepedia.org/galois-group) of an extension of local fields (or of the decomposition group at a prime of a number field) that measures how deeply each automorphism ramifies the extension. The first group, the inertia group, collects automorphisms acting trivially on the residue field; the higher groups in the filtration isolate the wild ramification, the part controlled by the residue characteristic. The theory, developed in its standard modern form in [Jean-Pierre Serre](https://www.edgepedia.org/jean-pierre-serre)'s *Local Fields* (Chapter IV), underlies the Artin conductor, the conductor-discriminant formula, and the study of the different.<sup>[1](https://www.math.columbia.edu/~chaoli/docs/ClassFieldTheory.html)</sup><sup> • </sup><sup>[3](https://www.lmfdb.org/knowledge/show/artin.conductor)</sup>

| Key fact | Detail |
|---|---|
| Definition (lower numbering) | For a finite Galois extension \( L/K \) of local fields with group \( G \), \( G_i = \ker(G \to \mathrm{Aut}(\mathcal{O}_L/\mathfrak{m}_L^{i+1})) \), giving \( G = G_{-1} \supseteq G_0 \supseteq G_1 \supseteq \cdots \supseteq \{1\} \)<sup>[1](https://www.math.columbia.edu/~chaoli/docs/ClassFieldTheory.html)</sup> |
| Inertia group | \( G_0 \) is the inertia group; \( \#G_0 = e(L/K) \), the ramification index, and \( L/L^{\mathrm{ur}} \) is totally ramified<sup>[1](https://www.math.columbia.edu/~chaoli/docs/ClassFieldTheory.html)</sup> |
| Tame and wild parts | \( G_0/G_1 \) is cyclic of order prime to \( p = \mathrm{Char}(k) \); \( G_1 \) is the unique \( p \)-Sylow subgroup of \( G_0 \), the wild inertia group<sup>[1](https://www.math.columbia.edu/~chaoli/docs/ClassFieldTheory.html)</sup> |
| Structure | For \( i \ge 1 \), \( G_i/G_{i+1} \) is an elementary \( p \)-group, so \( G \) is solvable<sup>[1](https://www.math.columbia.edu/~chaoli/docs/ClassFieldTheory.html)</sup> |
| Upper numbering | Via the Hasse-Herbrand function \( \phi(u) = \int_0^u dt/[G_0:G_t] \), one sets \( G^v = G_{\phi(v)} \); then \( G^{-1} = G \), \( G^0 = G_0 \), and \( G^v = 1 \) for \( v \gg 0 \)<sup>[1](https://www.math.columbia.edu/~chaoli/docs/ClassFieldTheory.html)</sup> |
| Quotient behavior | Upper numbering is preserved when the Galois group is replaced by a quotient; lower numbering is not<sup>[2](https://mathworld.wolfram.com/RamificationGroup.html)</sup> |
| Main application | The filtration in lower numbering defines the exponents \( f(\rho, \mathfrak{p}) \) in the Artin conductor of an Artin representation<sup>[3](https://www.lmfdb.org/knowledge/show/artin.conductor)</sup> |

## Lower numbering

Let \( L/K \) be a finite [Galois extension](https://www.edgepedia.org/galois-extension) of non-Archimedean local fields with Galois group \( G \), valuation ring \( \mathcal{O}_L \), and maximal ideal \( \mathfrak{m}_L \). For each integer \( i \ge 0 \), the \( i \)-th ramification group in the lower numbering is the subgroup of automorphisms acting trivially on the quotient \( \mathcal{O}_L/\mathfrak{m}_L^{i+1} \); equivalently, \( G_i \) is the kernel of the natural map \( G \to \mathrm{Aut}(\mathcal{O}_L/\mathfrak{m}_L^{i+1}) \). Together with the convention \( G_{-1} = G \), these subgroups form a chain

\[ G = G_{-1} \supseteq G_0 \supseteq G_1 \supseteq \cdots \supseteq \{1\}. \]

The filtration is finite: for a finite extension, \( G_i \) is trivial once \( i \) is large enough.<sup>[1](https://www.math.columbia.edu/~chaoli/docs/ClassFieldTheory.html)</sup>

The first two steps have direct field-theoretic meaning. The group \( G_0 \) is the inertia group, the kernel of the action on the residue field \( l/k \); it satisfies \( G/G_0 = \mathrm{Gal}(L^{\mathrm{ur}}/K) \) and \( G_0 = \mathrm{Gal}(L/L^{\mathrm{ur}}) \), where \( L^{\mathrm{ur}} \) is the maximal unramified subextension. Its order equals the ramification index \( e(L/K) \).<sup>[1](https://www.math.columbia.edu/~chaoli/docs/ClassFieldTheory.html)</sup>

**Tame versus wild ramification.** The fixed field of \( G_1 \) is the maximal tamely ramified subextension \( L^{\mathrm{t}} \). The quotient \( G_0/G_1 \cong \mathrm{Gal}(L^{\mathrm{t}}/L^{\mathrm{ur}}) \) is called the tame inertia group; it is cyclic of order prime to \( p \), the characteristic of the residue field. The subgroup \( G_1 \cong \mathrm{Gal}(L/L^{\mathrm{t}}) \) is the wild inertia group and is the unique \( p \)-Sylow subgroup of \( G_0 \). Thus the filtration separates the two kinds of ramification: tame ramification, which behaves like the classical theory of cyclic extensions, and wild ramification, which exists only in residue characteristic \( p \).<sup>[1](https://www.math.columbia.edu/~chaoli/docs/ClassFieldTheory.html)</sup>

The higher quotients are tightly constrained: for \( i \ge 1 \), \( G_i/G_{i+1} \) is an elementary abelian \( p \)-group. A consequence is that the Galois group \( G \) of any finite Galois extension of local fields is solvable.<sup>[1](https://www.math.columbia.edu/~chaoli/docs/ClassFieldTheory.html)</sup>

## Upper numbering and the Hasse-Herbrand function

The lower numbering behaves poorly under passage to subextensions or quotients, so Serre introduces a reparametrization. Define, for \( u \ge -1 \),

\[ \phi(u) = \int_0^u \frac{dt}{[G_0:G_t]}, \]

with the convention that for non-integer values of u one sets \( G_u = G_{\lceil u \rceil} \), while defining \( [G_0:G_{-1}] \) as the reciprocal of \( [G_{-1}:G_0] \) and taking \( [G_0:G_t] \) to equal 1 whenever \( t \in (-1,0] \). The function \( \phi \) is piecewise linear, continuous, increasing, and convex-up; its inverse is denoted \( \psi \). The ramification groups in the upper numbering are then \( G^v = G_{\phi(v)} \). This reindexing satisfies \( G^{-1} = G_{-1} = G \), \( G^0 = G_0 \), and \( G^v = 1 \) for \( v \) sufficiently large.<sup>[1](https://www.math.columbia.edu/~chaoli/docs/ClassFieldTheory.html)</sup> The variable \( v \) here is a real number, not to be confused with a valuation map.<sup>[4](https://math.arizona.edu/~cais/scans/Cassels-Frohlich-Algebraic_Number_Theory.pdf)</sup>

The point of the construction is functoriality. If \( L' \) is a Galois intermediate extension of \( L/K \), the restriction map \( \mathrm{Gal}(L/K) \to \mathrm{Gal}(L'/K) \) sends \( G_u(L/K) \) to \( G_v(L'/K) \) with \( v = \psi(u) \).<sup>[14](https://math.ucla.edu/~sharifi/notes/algnum-ch06.html)</sup> Herbrand's theorem states that for any normal subgroup \( H \lhd G \), identifying \( \mathrm{Gal}(L^H/K) \) with \( G/H \), one has \( (G/H)^v = G^v H/H \): the upper numbering is compatible with quotients. The lower numbering lacks this property.<sup>[1](https://www.math.columbia.edu/~chaoli/docs/ClassFieldTheory.html)</sup><sup> • </sup><sup>[2](https://mathworld.wolfram.com/RamificationGroup.html)</sup>

A number \( v \) at which the upper filtration jumps, that is, where \( G^v \neq G^{v+\epsilon} \) for every \( \epsilon > 0 \), is called a break in the filtration; the breaks encode the ramification data of the extension.<sup>[5](https://www.researchgate.net/publication/243117602_Ramification_theory_in_non-abelian_local_class_field_theory)</sup>

## Relation to the different and the discriminant

The filtration is not merely bookkeeping: it computes classical invariants of the extension. There is a formula for the valuation of the different ideal \( \mathfrak{D}_{L/K} \) in terms of the orders of the higher ramification groups, which links the group-theoretic filtration to the different, and hence to the discriminant of a number field.<sup>[6](https://math.stanford.edu/~conrad/papers/notes.pdf)</sup> In the opposite direction, ramification groups and Artin conductors have been computed explicitly for families of extensions, for example radical extensions of the rationals, where the conductor-discriminant formula then yields discriminant formulas.<sup>[7](https://arxiv.org/abs/math/0409533)</sup>

## Applications

**Artin conductor.** The principal application of the higher ramification filtration is the Artin conductor. For an Artin representation \( \rho \) of the absolute Galois group of a number field, the conductor is a positive integer measuring the ramification of \( \rho \). To define the local exponent \( f(\rho, \mathfrak{p}) \) at a prime \( \mathfrak{p} \), one takes the Galois group \( G \) of the corresponding local extension and uses its filtration \( G_i \) in the lower numbering, as defined in Chapter IV of Serre's *Local Fields*. The conductor enters the functional equation of the Artin \( L \)-function \( L(s, \chi) \).<sup>[3](https://www.lmfdb.org/knowledge/show/artin.conductor)</sup><sup> • </sup><sup>[8](https://sites.math.duke.edu/~dasgupta/papers/Dasguptaseniorthesis.pdf)</sup>

**Hasse-Arf and its descendants.** The Hasse-Arf theorem asserts integrality of the upper-numbering jumps for abelian representations. A modern outgrowth carries this idea forward: given an \( l \)-adic sheaf on a variety over a local field, the Swan class is defined as a 0-cycle class whose support lies in the wild ramification locus, and for curves over a local field its integrality is a generalization of the Hasse-Arf theorem; within this same framework one can also establish the conjecture Serre made about the Artin character when a group acts on a 2-dimensional regular local ring with a single fixed point.<sup>[9](https://arxiv.org/abs/1007.0310)</sup>

**Extensions of the theory.** The classical filtration assumes a perfect residue field. For local fields with imperfect residue fields, a modified ramification filtration has been constructed with a Hasse-Herbrand type function retaining the usual properties, and the resulting ramification groups have been characterized intrinsically.<sup>[10](https://arxiv.org/abs/math/0201238)</sup><sup> • </sup><sup>[11](https://arxiv.org/abs/2004.03770)</sup> The computation of ramification groups for specific classes of extensions remains active: cyclic extensions of degree \( p \), described by Artin-Schreier equations in characteristic \( p \) and Kummer equations in characteristic 0, admit explicit lower ramification groups and refined Swan conductors.<sup>[12](https://www.cambridge.org/core/books/abs/ramification-groups-of-local-fields/trace-of-differential-forms/C46AD2E8EAEF7F02338746D4A8F6207E)</sup> In 2025, Stephen DeBacker, David Schwein, and Cheng-Chiang Tsai introduced a normalized indexing of the lower and upper numbering ramification subgroups, arguing that it is better suited for studying base change for Langlands parameters.<sup>[13](https://arxiv.org/abs/2509.14881)</sup>

## References

1. [Class field theory, lecture notes of Chao Li, Columbia University (Definitions 27-29, Remarks 22-23)](https://www.math.columbia.edu/~chaoli/docs/ClassFieldTheory.html)
2. [Ramification Group, Wolfram MathWorld](https://mathworld.wolfram.com/RamificationGroup.html)
3. [Conductor of an Artin representation, LMFDB knowledge page](https://www.lmfdb.org/knowledge/show/artin.conductor)
4. [J.-P. Serre, Local Fields, in Cassels and Fröhlich (eds.), Algebraic Number Theory (scan)](https://math.arizona.edu/~cais/scans/Cassels-Frohlich-Algebraic_Number_Theory.pdf)
5. [Ramification theory in non-abelian local class field theory, ResearchGate](https://www.researchgate.net/publication/243117602_Ramification_theory_in_non-abelian_local_class_field_theory)
6. [Notes on local fields (different and higher ramification groups), Stanford](https://math.stanford.edu/~conrad/papers/notes.pdf)
7. [Ramification groups and Artin conductors of radical extensions of the rationals, arXiv:math/0409533](https://arxiv.org/abs/math/0409533)
8. [A.4 Statements from local class field theory, A. Dasgupta senior thesis](https://sites.math.duke.edu/~dasgupta/papers/Dasguptaseniorthesis.pdf)
9. [Ramification theory for varieties over a local field, arXiv:1007.0310](https://arxiv.org/abs/1007.0310)
10. [On ramification theory in the imperfect residue field case, arXiv:math/0201238](https://arxiv.org/abs/math/0201238)
11. [A characterization of ramification groups of local fields with imperfect residue field, arXiv:2004.03770](https://arxiv.org/abs/2004.03770)
12. [Trace of Differential Forms, Chapter 4 of Ramification Groups of Local Fields, Cambridge University Press](https://www.cambridge.org/core/books/abs/ramification-groups-of-local-fields/trace-of-differential-forms/C46AD2E8EAEF7F02338746D4A8F6207E)
13. [Stephen DeBacker, David Schwein, Cheng-Chiang Tsai, Normalized Indexing for Ramification Subgroups, arXiv:2509.14881](https://arxiv.org/abs/2509.14881)
14. [Algebraic Number Theory, Chapter 6 Ramification theory · Romyar Sharifi](https://math.ucla.edu/~sharifi/notes/algnum-ch06.html)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Number theory › Algebraic number theory › Class field theory › Local class field theory*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

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