Fact-checked by Grok 2 weeks ago

Ramification group

In algebraic number theory, ramification groups are a decreasing filtration of normal subgroups of the Galois group of a finite Galois extension of local fields with discrete valuation, designed to measure the degree and nature of ramification at a prime ideal.^[1] For a Galois extension L/K of complete discrete valuation fields with valuation rings \mathcal{O}_K and \mathcal{O}_L, and uniformizers \pi_K and \pi_L, the i-th lower ramification group G_i (for i \geq -1) consists of those \sigma \in \mathrm{Gal}(L/K) such that v_L(\sigma(\alpha) - \alpha) \geq i+1 for all \alpha \in \mathcal{O}_L, where v_L is the L-adic valuation normalized so that v_L(\pi_L) = 1; here, G_{-1} is the full Galois group, G_0 is the inertia group capturing tame ramification, and higher G_i for i \geq 1 detect wild ramification.^[2]^[3] This filtration G = G_{-1} \supseteq G_0 \supseteq G_1 \supseteq \cdots terminates at the trivial subgroup for sufficiently large i, providing a refined structure on the decomposition group at the prime, which itself is the stabilizer of a prime ideal \mathfrak{p}_L above \mathfrak{p}_K in the global setting of number fields.^[1] The ramification index e = e(L/K) = [L:K]_{\mathfrak{p}_L} / f, where f is the residue degree, relates directly to these groups: the extension is unramified if G_0 = \{1\}, tamely ramified if the wild inertia subgroup G_1 (the p-Sylow subgroup of G_0) is trivial, and wildly ramified otherwise, with G_1 often a p-group in characteristic p > 0 or mixed in positive characteristic.^[3]^[2] Higher ramification groups extend this via upper numbering, using the Herbrand function \phi(u) = \int_0^u \frac{dt}{g(t)} (where g(t) = |G_0 : G_t|) to define G^u = G_{\psi(u)} for real u \geq -1, ensuring compatibility under quotients and enabling precise control in infinite extensions or towers.^[1] The valuation of the different ideal \mathfrak{d}_{L/K} is given by v_{\mathfrak{p}_L}(\mathfrak{d}_{L/K}) = \sum_{i=0}^\infty (|G_i| - 1), quantifying total ramification and linking local behavior to global arithmetic invariants like the discriminant.^[3] Ramification groups play a foundational role in local class field theory, where they describe the kernel of the Artin reciprocity map and bound the structure of abelian extensions, as well as in the Hasse-Arf theorem, which asserts that jumps in the filtration occur at integers for abelian cases.^[2] They also appear in geometric contexts, such as analyzing ramification loci in étale covers of schemes or Puiseux expansions in algebraic closures of Laurent series fields.^[1] Developed in the early 20th century through works on local fields, these groups remain essential for inverse Galois problems, effective computation of class numbers, and understanding ramification in global fields via completions.^[1]

Preliminaries in local number theory

Local fields and discrete valuations

A local field is a field that is complete with respect to a discrete valuation and has a finite residue field.^[4] Examples include the field of p-adic numbers \mathbb{Q}_p for a prime p, which is the completion of \mathbb{Q} with respect to the p-adic valuation, and finite extensions of the field of formal Laurent series \mathbb{F}_p((t)) over the finite field \mathbb{F}_p. These fields provide the local setting for studying ramification in number theory and algebraic geometry.^[5] A discrete valuation on a field K is a surjective group homomorphism v: K^\times \to \mathbb{Z} satisfying v(xy) = v(x) + v(y) for all x, y \in K^\times, extended by v(0) = \infty.^[4] It is normalized such that v(p) = 1 in the case of \mathbb{Q}_p. The associated valuation ring is O_K = \{ x \in K \mid v(x) \geq 0 \} \cup \{0\}, which is a discrete valuation ring, and the maximal ideal is m_K = \{ x \in K \mid v(x) > 0 \}.^[5] The residue field is the finite field k_K = O_K / m_K.^[4] A uniformizer \pi_K \in O_K is an element with v(\pi_K) = 1, generating m_K as m_K = \pi_K O_K. The valuation induces a topology on K via the metric d(x,y) = c^{-v(x-y)} for some c > 1, making K locally compact and Hausdorff when complete.^[5] For a finite extension L/K of local fields, the discrete valuation on K extends uniquely to a discrete valuation on L, yielding the valuation ring O_L and uniformizer \pi_L with v_L(\pi_L) = 1.^[4] The ring O_L is the integral closure of O_K in L, and L remains complete with respect to this extended valuation.

Galois extensions and ramification indices

In the Galois-theoretic framework for extensions of local fields, a finite Galois extension L/K is equipped with the Galois group G = \mathrm{Gal}(L/K). Since K is a non-archimedean local field with ring of integers O_K and unique maximal ideal \mathfrak{m}_K, the integral closure O_L in L is also the ring of integers of L, which is a discrete valuation ring with a unique maximal ideal \mathfrak{m}_L lying above \mathfrak{m}_K, satisfying \mathfrak{m}_K O_L = \mathfrak{m}_L^{e(L/K)}.^[6] The ramification index e(L/K) is the positive integer e such that if \pi_K is a uniformizer of K (i.e., a generator of \mathfrak{m}_K), then the extended valuation satisfies v_L(\pi_K) = e.^[7] Equivalently, e(L/K) measures the exponent to which \mathfrak{m}_L appears in the factorization of \mathfrak{m}_K O_L. The residue degree f(L/K) is defined as the degree of the extension of residue fields [k_L : k_K], where k_K = O_K / \mathfrak{m}_K and k_L = O_L / \mathfrak{m}_L. For any finite extension L/K of local fields, the fundamental relation [L : K] = e(L/K) f(L/K) holds, reflecting the decomposition of the degree into ramification and inertial components.^[6]^[7] An extension L/K is unramified if and only if e(L/K) = 1, in which case f(L/K) = [L : K] and the residue field extension k_L / k_K determines L/K uniquely as the (Henselian) lift of this residue extension to characteristic zero.^[8] Conversely, L/K is totally ramified if f(L/K) = 1, so the residue fields are isomorphic (k_L \cong k_K) and e(L/K) = [L : K], with the extension arising purely from valuation considerations without inertial growth.^[9] Ramification in local field extensions is further classified as tame or wild relative to the characteristic p > 0 of the residue field k_K. The extension L/K is tamely ramified if p does not divide e(L/K), ensuring that the ramification is controlled by roots of unity or cyclic actions of order coprime to p. Otherwise, if p divides e(L/K), the extension is wildly ramified, leading to more intricate behavior involving p-power structures in the extension.^[10]^[6]

Fundamental ramification subgroups

Decomposition group

In a finite Galois extension L/K of number fields, with Galois group G = \Gal(L/K), let \mathfrak{p} be a prime ideal of the ring of integers \mathcal{O}_K and \mathfrak{w} a prime ideal of \mathcal{O}_L lying above \mathfrak{p}. The decomposition group D_{\mathfrak{w}} at \mathfrak{w} is the stabilizer subgroup D_{\mathfrak{w}} = \{\sigma \in G \mid \sigma(\mathfrak{w}) = \mathfrak{w}\}.^[11] This subgroup is closed under conjugation and all decomposition groups at primes above \mathfrak{p} are conjugate in G.^[2] The index [G : D_{\mathfrak{w}}] equals the number g of distinct prime ideals of \mathcal{O}_L lying above \mathfrak{p}, reflecting the transitive action of G on these primes via the orbit-stabilizer theorem.^[11] The decomposition group D_{\mathfrak{w}} admits a canonical isomorphism with the Galois group of the corresponding local extension obtained by completion. Specifically, D_{\mathfrak{w}} \cong \Gal(L_{\mathfrak{w}}/K_{\mathfrak{p}}), where L_{\mathfrak{w}} is the completion of L at \mathfrak{w} and K_{\mathfrak{p}} is the completion of K at \mathfrak{p}.^[2] This isomorphism arises from the natural embedding of the global Galois action into the local one, preserving the decomposition of the maximal ideal. The kernel of the induced surjection D_{\mathfrak{w}} \to \Gal(k_{\mathfrak{w}}/k_{\mathfrak{p}}), where k_{\mathfrak{w}} and k_{\mathfrak{p}} are the residue fields, is the inertia subgroup I_{\mathfrak{w}}, which captures the ramification.^[11] When L/K is already a finite Galois extension of complete discrete valuation fields (local fields) with respect to uniformizers and maximal ideals \mathfrak{m}_L \subset \mathcal{O}_L and \mathfrak{m}_K \subset \mathcal{O}_K, the decomposition group coincides with the full Galois group D = G = \Gal(L/K). In this local setting, G acts on the residue field extension k_L / k_K via reduction modulo \mathfrak{m}_L, yielding a surjective homomorphism G \to \Aut(k_L / k_K) whose kernel is the inertia subgroup.^[2]

Inertia group

In the context of a Galois extension L_w / K of local fields, where w lies over the prime \mathfrak{p} of K, the inertia group I_w is defined as the kernel of the action of the decomposition group D_w on the residue field k_L of L_w. Explicitly, I_w = \{\sigma \in D_w \mid \sigma(x) \equiv x \pmod{\mathfrak{m}_L} \ \forall x \in \mathcal{O}_L \}, where \mathcal{O}_L is the ring of integers of L_w and \mathfrak{m}_L its maximal ideal.^[1] This subgroup consists of those elements in D_w that act trivially on the residue field, thereby capturing the ramification behavior of the extension. The inertia group I_w is a normal subgroup of D_w, and the quotient D_w / I_w is isomorphic to the Galois group \mathrm{Gal}(k_L / k_K) of the residue field extension. Consequently, the order of I_w equals the ramification index e(L_w / K), reflecting the degree to which the valuation ramifies in the extension. This isomorphism highlights how the inertia group isolates the ramification from the inertial (residue) part of the decomposition group. The inertia group further decomposes into tame and wild components. The wild inertia subgroup P_w is the maximal pro-p-subgroup of I_w, where p is the residue characteristic, and it coincides with the first higher ramification group G_1. The tame inertia is then the quotient I_w / P_w, which acts faithfully on the roots of unity of order prime to p in the extension.^[1] This distinction separates mildly ramified (tame) extensions from those involving higher p-powers (wild). The extension L_w / K is unramified if and only if I_w is trivial, meaning the ramification index e(L_w / K) = 1. In this case, the decomposition group D_w acts solely through the residue field Galois group, with no valuation distortion.

Ramification groups in lower numbering

Definition and filtration

In the context of a finite Galois extension L/K of complete discrete valuation fields with residue characteristic p, the higher ramification groups in lower numbering provide a decreasing filtration of the inertia group, refining the distinction between tame and wild ramification.^[12] For each integer i \geq 0, the i-th lower ramification group G_i (also denoted G_{v,i} where v = v_L is the valuation on L) is defined as the subgroup

G_i = \{\sigma \in G \mid v(\sigma(\alpha) - \alpha) \geq i + 1 \text{ for all } \alpha \in \mathcal{O}_L \},

where G = \mathrm{Gal}(L/K) and \mathcal{O}_L is the ring of integers of L.^[13]^[12] This condition means that \sigma acts trivially on \mathcal{O}_L modulo the (i+1)-th power of the maximal ideal \mathfrak{m}_L.^[14] These groups form a decreasing filtration G_0 \supseteq G_1 \supseteq G_2 \supseteq \cdots of the inertia group, with G_0 = I the full inertia subgroup and G_1 = P the wild inertia subgroup, which is a pro-p group.^[14]^[12] Each G_i is normal in G, and the intersection \bigcap_{i \geq 0} G_i = \{1\}.^[13]^[12] The points of ramification, or jumps, occur at those integers i \geq 0 where G_i \ properly\supset G_{i+1}, marking the breaks in the filtration where the quotients G_i / G_{i+1} are nontrivial.^[12]

Properties and the different ideal

The lower ramification filtration exhibits transitivity with respect to intermediate extensions. Specifically, for a Galois extension L/K of local fields with an intermediate field M such that L/M/K, the ramification groups of \mathrm{Gal}(L/M) are given by \mathrm{Gal}(L/M) \cap G_i for each i \geq 0, where G_i are the lower ramification groups of \mathrm{Gal}(L/K).^[1] The ramification groups play a central role in determining the different ideal \mathcal{D}_{L/K}, which quantifies the ramification in the extension. For a finite Galois extension L/K, the valuation of the different is given by

v_L(\mathcal{D}_{L/K}) = \sum_{i=0}^\infty (|G_i| - 1),

where the sum is finite since G_i = \{1\} for sufficiently large i.^[15] The different ideal serves as a measure of ramification depth and is instrumental in computing the discriminant ideal of the extension, which in turn relates to the conductor in local class field theory.^[15]

Examples of lower ramification groups

Cyclotomic extensions

The cyclotomic extension L = \mathbb{Q}_p(\zeta_{p^n}) over K = \mathbb{Q}_p is a totally ramified Galois extension of degree \phi(p^n) = p^{n-1}(p-1), with Galois group isomorphic to (\mathbb{Z}/p^n\mathbb{Z})^\times. For p odd, this group decomposes as a direct product \mathbb{Z}/(p-1)\mathbb{Z} \times \mathbb{Z}/p^{n-1}\mathbb{Z}, reflecting the tame and wild ramification components. The extension is abelian, and the decomposition group coincides with the full Galois group G, as there is no unramified part. The inertia group I = G_0 equals the entire Galois group G, confirming total ramification with no residue field extension. The first higher ramification group G_1 is the Sylow p-subgroup of G, known as the wild inertia subgroup, isomorphic to \mathbb{Z}/p^{n-1}\mathbb{Z}. This subgroup corresponds to the kernel of the natural projection (\mathbb{Z}/p^n\mathbb{Z})^\times \to (\mathbb{Z}/p\mathbb{Z})^\times, and it can be identified with the quotient of principal units $1 + p\mathbb{Z}_p / (1 + p^n \mathbb{Z}_p).^[4] For i > 0, the higher ramification groups G_i are subgroups of G_1, forming a filtration on the pro-p wild inertia. These groups are determined explicitly by the action of Galois elements on a uniformizer \pi = \zeta_{p^n} - 1 of L, where \sigma \in G_i if and only if v_L(\sigma(\pi) - \pi) \geq i + 1. The elements of G act via \sigma(\zeta_{p^n}) = \zeta_{p^n}^m for m \in (\mathbb{Z}/p^n\mathbb{Z})^\times, and the valuation v_L(\sigma(\pi) - \pi) depends on the p-adic order of m - 1. Specifically, the filtration jumps at i = 1, with subsequent structure scaled by the tame factor p-1; in particular, G_i \cong 1 + p^k \mathbb{Z}_p / (1 + p^n \mathbb{Z}_p) for appropriate k depending on i, where the groups stabilize in intervals until G_i = \{1\} for sufficiently large i < p^{n-1}(p-1). This explicit description illustrates the abelian totally ramified case, with the pro-p structure providing a prototypical example of wild ramification.^[4] The different ideal \mathcal{D}_{L/K} satisfies v_L(\mathcal{D}_{L/K}) = \sum_{i \geq 0} (|G_i| - 1), which quantifies the cumulative effect of tame and wild ramification in the tower and decreases relative to the degree as n increases due to the deepening pro-p structure.

Quartic extensions

A standard example of a quartic Galois extension exhibiting wild ramification is L = \mathbb{Q}_2(\zeta_8)/K = \mathbb{Q}_2, where \zeta_8 is a primitive 8th root of unity satisfying the minimal polynomial x^4 + 1 = 0. This extension has degree 4 and Galois group isomorphic to the Klein four-group V_4 = \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}. The extension is totally ramified with ramification index e = 4 and residue degree f = 1, so the unique prime of K lies below the unique prime of L with ramification index 4. Consequently, the decomposition group D is the full Galois group G of order 4, and the inertia group I = G_0 = G of order 4. The wild nature of the ramification is evident since the prime p = 2 divides e = 4. The higher lower ramification groups are G_1 of order 2 and G_2 = 1. To see this explicitly, a uniformizer \pi of L can be taken as \pi = \zeta_8 - 1, and using the normalized valuation v_L with v_L(\pi) = 1 (so v_L(2) = 4), the Galois group acts on \zeta_8 by \sigma_k(\zeta_8) = \zeta_8^{1 + 2k} for k = 0,1,2,3, corresponding to the units mod 8. The subgroup G_1 consists of those \sigma_k with v_L(\sigma_k(\pi) - \pi) \geq 2, which holds for the index 2 subgroup generated by the automorphism sending \zeta_8 to \zeta_8^5 = -\zeta_8 (order 2 element fixing \zeta_8^2 = i), while the full G_0 acts with valuations exceeding 1 but not all exceeding 2, leading to G_2 = 1. This filtration shows a jump at i = 1, characteristic of wild ramification. The different ideal \mathcal{D}_{L/K} has valuation v_L(\mathcal{D}_{L/K}) = \sum_{i=0}^\infty (|G_i| - 1) = (4 - 1) + (2 - 1) = 4, which exceeds the tame bound e - 1 = 3 by 1, indicating the wild contribution through the nontrivial G_1. Using properties of lower ramification groups, the different highlights the non-trivial 2-Sylow subgroup in the inertia.

Ramification groups in upper numbering

Definition and transformation from lower numbering

The upper numbering ramification groups arise from a continuous transformation of the lower ramification filtration \{G_v\}_{v \geq 0}, designed to facilitate analysis under quotients of the Galois group G.^[1] The Herbrand function \phi: \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0} is defined by

\phi(v) = \int_0^v \frac{dt}{|G_0 : G_t|},

where the lower ramification groups G_t for non-integer t are taken as G_{\lfloor t \rfloor} in intervals of constancy, ensuring the integral is well-defined and piecewise linear.^[1] This function \phi is continuous and strictly increasing, serving as a homeomorphism with \phi(0) = 0 and \lim_{v \to \infty} \phi(v) = \infty.^[1] Its inverse \psi = \phi^{-1}: \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0} is likewise a continuous, strictly increasing homeomorphism.^[1] The upper ramification groups are then defined as G^u = G_{\psi(u)} for all u \geq 0.^[1] Consequently, G^0 = G_{\psi(0)} = G_0, and G^\infty = \bigcap_{u \geq 0} G^u = \{1\}, reflecting the eventual triviality of the filtration.^[1] The map u \mapsto G^u yields a decreasing filtration, with u < u' implying G^{u'} \subseteq G^u.^[1] This transformation preserves normality and compatibility with quotients: if H \trianglelefteq G, then the upper ramification groups of G/H satisfy (G/H)^u = G^u H / H for all u \geq 0.^[1] Such properties make the upper numbering particularly suited for quotient structures, smoothing irregularities from non-integral jumps in the lower filtration by rescaling via the varying indices |G_0 : G_t|.^[1]

Herbrand's theorem

Herbrand's theorem asserts that the upper numbering ramification groups behave compatibly with quotients of the Galois group. Specifically, let G = \Gal(L/K) be the Galois group of a finite Galois extension of local fields, and let H \trianglelefteq G be a normal subgroup with fixed field F. Then, for every real number v \geq -1, the upper ramification group (G/H)^v of the subextension F/K is the image G^v H / H, where G^v denotes the v-th upper ramification group of L/K. This property establishes the upper numbering as the canonical refinement of the ramification filtration. In particular, for $0 \leq u \leq u', the quotient G^u / G^{u'} is canonically isomorphic to the upper ramification group of the intermediate extension corresponding to the fixed field of G^{u'} relative to the fixed field of G^u. Moreover, the upper ramification groups form a decreasing filtration G^0 \supseteq G^u \supseteq G^{u'} \supseteq \cdots for $0 \leq u \leq u', with G^u constant between jumps and containing the wild inertia group G^1. A proof sketch proceeds via the Herbrand functions \phi_{L/K} and \psi_{L/K}, which relate the lower and upper numberings. Recall that \phi_{L/K}(u) = \int_0^u \frac{dt}{[G_0 : G_t]} for u \geq 0, where G_t are the lower ramification groups (and extended appropriately for -1 \leq u < 0), and \psi_{L/K} is its continuous, strictly increasing inverse, with upper groups defined by G^v = G_{\psi_{L/K}(v)}. The compatibility follows from the transitivity of Herbrand functions: \phi_{L/K} = \phi_{L/F} \circ \phi_{F/K}. This implies that the indices satisfy [G_0 : G^v] = [(G/H)_0 : (G/H)^v] \cdot [H_0 : H^v], leading to (G/H)^v = G^v H / H. As a corollary, the upper numbering is independent of auxiliary choices, such as the selection of finite quotients in infinite Galois extensions, enabling a consistent definition for profinite groups like absolute Galois groups of local fields. Additionally, the locations of jumps (discontinuities) in the lower numbering filtration transform under the \psi function to yield the jumps in the upper numbering. The upper numbering further refines the structure of the different ideal \mathfrak{D}_{L/K}. Since the Herbrand functions compose in towers—specifically, \phi_{L/K} = \phi_{L/F} \circ \phi_{F/K}—the valuation v(\mathfrak{D}_{L/K}) = v(\mathfrak{D}_{L/F}) + v(\mathfrak{D}_{F/K}) can be analyzed additively using the upper filtration, providing finer control over ramification contributions in composite extensions compared to the lower numbering.^[1]

Advanced results and applications

Hasse-Arf theorem

The Hasse–Arf theorem asserts that for a finite abelian extension L/K of local fields (complete with respect to a discrete valuation and with finite residue field), the jumps in the upper ramification filtration of the Galois group G = \mathrm{Gal}(L/K) occur at integer values. Specifically, if G^u \neq G^{u+\epsilon} for all \epsilon > 0, where G^u denotes the u-th upper ramification group, then u is an integer.^[16] This filtration is obtained by reindexing the lower ramification groups via the Herbrand function \phi(u) = \int_0^u \frac{dt}{|G_0 : G_t|}, which transforms the lower jumps into upper ones, preserving the structure but normalizing the breaks for composita.^[17] The proof proceeds via local class field theory, which establishes a bijection between finite abelian extensions of K and open subgroups of finite index in the multiplicative group K^\times that are norm subgroups N_{L/K} L^\times. Under this correspondence, the upper ramification groups G^u align with the subgroups U_n of higher unit groups (principal units congruent to 1 modulo \mathfrak{p}_K^{n+1}, where \mathfrak{p}_K is the maximal ideal of the ring of integers of K), specifically G^u = \Psi_{L/K} (N_{L/K} U_{\lceil u \rceil, L}) for the reciprocity map \Psi_{L/K} : K^\times / N_{L/K} L^\times \to G. The integrality of jumps then follows because the conductor of the extension—the minimal f such that G^u = 1 for u \geq f—is an integer exponent tied to the Artin conductor, ensuring that breaks in the filtration correspond to integer levels in the unit group hierarchy.^[17]^[18] In contrast, non-abelian extensions may exhibit non-integer jumps in the upper numbering, highlighting the theorem's specialization to abelian cases. For instance, consider a totally ramified extension L/K of local fields with Galois group the quaternion group Q_8 = \{\pm 1, \pm i, \pm j, \pm k\}; here, the lower ramification groups satisfy G_0 = G_1 = G_2 = G_3 = Q_8 and G_4 = 1, but the upper jumps occur at u = 1 (full drop to the center Z(Q_8) = \{\pm 1\}) and u = 3/2 (drop to trivial), violating integrality.^[19] This integrality simplifies explicit computations of ramification in abelian settings, such as cyclotomic extensions K(\zeta_{p^n})/K for prime p, where the jumps align with known integer conductor exponents (e.g., at u = 1, 2, \dots, n for totally ramified p-parts), enabling precise determination of inertia and wild ramification without fractional adjustments.^[16]

Connections to class field theory

In local class field theory, the Artin reciprocity map provides a canonical isomorphism between the multiplicative group of the base field and the Galois group of its maximal abelian extension. For a finite abelian extension L/K of non-archimedean local fields, the map \theta_{L/K}: K^\times / N_{L/K} L^\times \to \Gal(L/K) is an isomorphism, where N_{L/K} L^\times is the norm subgroup.^[20] The higher ramification groups G^u of \Gal(L/K) correspond to the filtration on the unit group U_K via the local Artin map, identifying open subgroups of finite index in the norm group with fixed fields of these ramification subgroups.^[21] This structure ensures that the reciprocity map encodes the ramification behavior through the inertia and wild inertia subgroups, distinguishing tame ramification (where G^1 = 1) from wild ramification (where higher groups G^u \neq 1 for u > 1).^[20] The conductor of the extension L/K is defined as the smallest integer f such that the higher unit group U_K(f) \subseteq N_{L/K} L^\times, measuring the extent of ramification.^[20] This conductor relates directly to the ramification groups via the Herbrand function \psi, where f is determined by the largest u such that G^u \neq 1, ensuring that the reciprocity map factors through the ray class group modulo the conductor.^[21] In the abelian case, the ramification groups thus classify the "tame" and "wild" components of the norm residue symbol, with tame ramification corresponding to quotients by the inertia group and wild ramification involving the p-Sylow structure of higher groups.^[20] On the global level, local ramification data from completions at finite primes controls the overall discriminant of an abelian extension L/K of number fields via the conductor-discriminant formula, where \disc(L/K) = \prod_\mathfrak{f} \mathfrak{f}^{n_\mathfrak{f}} with exponents tied to local conductors and ramification indices.^[22] This local-global linkage also influences class number formulas, as the Artin map's kernel incorporates local norm groups, relating the class number of K to the degree of the Hilbert class field through unramified local extensions at all primes.^[22]

References

[1]
[PDF] An Introduction to Higher Ramification Groups - UChicago Math
Definition 2. For any i ≥ −1, the ith ramification group Gi is defined to be. Gi = {σ ∈ G | vL(σ(α) − α) > i for all α ∈ AL}. Note that for any σ ∈ G, we have ...
[2]
[PDF] Math 223a: Algebraic Number Theory
The s-th ramification group of a prime P ⊆ OL is defined to be. Is(P∣p) ∶= {σ ∈ D(P∣p) ∶ σ(a) ≡ a mod Ps+1 for all a ∈ OL}, in other words it is the set ...
[3]
[PDF] Introduction to Algebraic Number Theory Lecture 14
Feb 14, 2014 · These are called higher ramification groups. The group V1 is called the “wild inertia” group and is denoted Pq/p. Theorem 6. Suppose L/K, q | ...
[4]
Local Fields | SpringerLink
This theory is about extensions-primarily abelian-of "local" (i.e., complete for a discrete valuation) fields with finite residue field. For example, such ...
[5]
[PDF] 9 Local fields and Hensel's lemmas
Oct 6, 2021 · In this lecture we introduce the notion of a local field; these are precisely the fields that arise as completions of a global field (finite ...
[6]
[PDF] Local Fields - Lecture Notes - Berkeley Math
Definition. A ring R is called a discrete valuation ring (DVR) if R is a principal ideal domain with exactly one non-zero prime ideal.
[7]
15.113 Galois extensions and ramification - Stacks Project
15.113 Galois extensions and ramification. In the case of Galois extensions, we can elaborate on the discussion in Section 15.112. Lemma 15.113.1.
[8]
[PDF] 11 Totally ramified extensions and Krasner's lemma
Oct 16, 2017 · ... Galois extension of K. In the example we were able to adjust our choice of the global field K without changing the local fields extension L/.<|control11|><|separator|>
[9]
[PDF] 1 Unramified Extensions 2 Totally Ramified Extensions - Arizona Math
Definition 1.1. An extension L/K of local fields is unramified if [L : K] = [l : k] with l = 0L/πL and. K = 0K/πK where πL,πK are uniformizers of L, ...
[10]
4.2 Cohomology of local fields: some computations - Kiran S. Kedlaya
Any finite Galois extension of local fields is solvable. To wit, the maximal unramified extension is cyclic,; the maximal tamely ramified extension is cyclic ...
[11]
[PDF] Math 129: Algebraic Number Theory Lecture 13: Galois Extensions
Mar 8, 2004 · The decomposition group is extremely useful because it allows us to see the extension K/Q as a tower of extensions, such that at each step in ...
[12]
[PDF] Math 676. Higher ramification groups Let K be complete with respect ...
filtration), see the discussion of ramification theory in Serre's Local Fields and Chapter 1 in the book edited ... inertia subgroup, its fixed field Kun has ...
[13]
[PDF] Math 845 Notes Class field theory
Definition 11.1 (Lower ramification groups). Let K be a nonarchimedean ... Neukirch, Algebraic Number Theory. [S] J.P. Serre, Local Fields. [Sil] J ...
[14]
[PDF] Local Fields - Tartarus
Jul 24, 2018 · Definition 2.2. A discrete valuation ring (DVR) is a PID with exactly one non-zero prime ideal (= with one maximal ideal). Lemma ...
[15]
[PDF] 18.785 Number Theory Fall 2017 Problem Set #10 Due
Dec 4, 2017 · The group Gi is the ith ramification group of G (in the lower ... [1] Jürgen Neukirch, Algebraic number theory, Springer-Verlag, 1999.<|control11|><|separator|>
[16]
[PDF] The different and differentials for local fields with imperfect residue ...
terms of the ramification groups by a well-known formula of Hilbert. ... Key words: local fields, ramification groups, different, differentials. ... For any local ...
[17]
4.4 Ramification filtrations and local reciprocity
We now introduce Herbrand's recipe to convert the lower numbering used in the definition of the ramification filtration into an upper numbering that behaves ...
[18]
[PDF] Class Field Theory
Class field theory describes the abelian extensions of a local or global field in terms of the arithmetic of the field itself. These notes contain an ...
[19]
[PDF] Hasse-Arf property and abelian extensions - Ivan Fesenko
We show how it easily follows from class field theory. Theorem (Hasse–Arf). Let L/F be a totally ramified abelian p-extension. Then. L/F satisfies HAP ...
[20]
[PDF] math 223a: upper numbering ramification groups
First, since L/K is totally ramified we must have G = G0. Second, recall from class that G0/G1 injects into the units of the residue field of K, so is cyclic.
[21]
[PDF] Class Field Theory
Class field theory describes the abelian extensions of a local or global field in terms of the arithmetic of the field itself.
[22]
[PDF] 27 Local class field theory
Dec 6, 2021 · We can extend the Artin map to K× by defining ψL/K(x) := ψL/K((x)); this map sends every uniformizer π to the. Frobenius element FrobL/K; note ...
[23]
[PDF] History of Class Field Theory - Mathematics
Hilbert starts with an abelian extension L/Q and uses his recently developed theory of higher ramification groups to show L lies in a succession of fields of ...