12.3 Decomposition and inertia groups

Decomposition and inertia groups are key players in understanding prime behavior in Galois extensions. They measure splitting and ramification, giving us insight into how primes behave when we extend our number fields.

These groups are essential tools in ramification theory, connecting local and global perspectives. By studying their structure and relationships, we can unravel the mysteries of prime ideal factorization and residue field extensions in algebraic number theory.

Decomposition and Inertia Groups

Definitions and Basic Properties

• Decomposition group D(P|p) of a prime ideal P in Galois extension L/K fixes P and measures splitting behavior
• Inertia group I(P|p) fixes the residue field extension and measures ramification
• D(P|p) acts transitively on prime ideals of L lying over p
• Fixed field of D(P|p) corresponds to decomposition field of extension for prime p
• Fixed field of I(P|p) corresponds to inertia field of extension for prime p
• Decomposition group contains both ramification and splitting information
• Inertia group focuses solely on ramification

Examples and Applications

• In quadratic extension $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$, prime 2 ramifies with D(P|2) = I(P|2) = Gal($\mathbb{Q}(\sqrt{2})/\mathbb{Q}$)
• For extension $\mathbb{Q}(\zeta_7)/\mathbb{Q}$ with $\zeta_7$ a 7th root of unity, prime 7 is totally ramified
• In cyclotomic extension $\mathbb{Q}(\zeta_p)/\mathbb{Q}$ for prime p, D(P|p) is cyclic of order p-1
• Local field extension $\mathbb{Q}_p(\sqrt[n]{p})/\mathbb{Q}_p$ has D(P|p) = Gal($\mathbb{Q}_p(\sqrt[n]{p})/\mathbb{Q}_p$)

Relationship of Decomposition and Inertia Groups

Group Structure and Quotients

• I(P|p) normal subgroup of D(P|p)
• Quotient D(P|p)/I(P|p) isomorphic to Galois group of residue field extension
• Index [D(P|p) : I(P|p)] equals degree of residue field extension
• Order |D(P|p)| product of ramification index, residue degree, and degree of splitting field
• Order |I(P|p)| equals ramification index of p in L/K
• Relationship expressed as |D(P|p)| = e(P|p) · f(P|p) · g(P|p) (e ramification index, f residue degree, g number of prime ideals over p)

Examples of Group Relationships

• Unramified prime in $\mathbb{Q}(\sqrt{5})/\mathbb{Q}$: I(P|p) trivial, D(P|p) cyclic of order 2
• Totally ramified prime in $\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$: D(P|2) = I(P|2) of order 3
• Split prime in $\mathbb{Q}(\sqrt{-1})/\mathbb{Q}$: D(P|p) trivial, I(P|p) trivial
• Tamely ramified prime in $\mathbb{Q}(\zeta_p)/\mathbb{Q}$: I(P|p) cyclic of order p-1

Structure of Decomposition and Inertia Groups

Specific Extension Types

• Unramified primes inertia group trivial, decomposition group cyclic (Frobenius automorphism generator)
• Totally ramified extensions decomposition and inertia groups coincide, order equals ramification index
• Tamely ramified extensions inertia group cyclic, order equals ramification index
• Wildly ramified extensions inertia group complex structure, often p-groups (p characteristic of residue field)
• Cyclotomic extensions analyzable using cyclotomic field theory and ramification behavior
• Local field extensions decomposition group entire Galois group, inertia group structure related to local class field theory

Examples in Various Extensions

• $\mathbb{Q}(\sqrt{-5})/\mathbb{Q}$: prime 2 inert, D(P|2) cyclic of order 2, I(P|2) trivial
• $\mathbb{Q}(\zeta_8)/\mathbb{Q}$: prime 2 totally ramified, D(P|2) = I(P|2) of order 4
• $\mathbb{Q}_p(\zeta_p)/\mathbb{Q}_p$: D(P|p) = Gal($\mathbb{Q}_p(\zeta_p)/\mathbb{Q}_p$), I(P|p) of order p-1
• $\mathbb{F}_p((t))(t^{1/p})/\mathbb{F}_p((t))$: wildly ramified, I(P|p) elementary abelian p-group

Ramification in Galois Extensions

Analysis Tools and Applications

• Decomposition and inertia groups analyze ramification of prime ideals in Galois extensions
• Ramification index of prime p in L/K equals order of inertia group I(P|p)
• Prime p splitting in L/K determined by decomposition group D(P|p) structure
• Higher ramification groups (subgroups of inertia group) provide wildly ramified extension depth information
• Hilbert different and discriminant computable using inertia groups of ramified primes
• Groups crucial in Hilbert-Speiser theorem formulation and proof for tamely ramified abelian extensions
• Essential for understanding prime behavior in tower extensions and ramification-Galois theory relationship

Concrete Examples and Computations

• $\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$: prime 2 ramification index 3, |I(P|2)| = 3
• $\mathbb{Q}(\zeta_7)/\mathbb{Q}$: prime 7 totally ramified, ramification index 6, |I(P|7)| = 6
• $\mathbb{Q}_2(\sqrt{2})/\mathbb{Q}_2$: wildly ramified, higher ramification groups G₀ = G₁ = I(P|2), G₂ = {1}
• Cyclotomic extension $\mathbb{Q}(\zeta_{p^n})/\mathbb{Q}$: prime p ramification index $p^{n-1}(p-1)$, |I(P|p)| = $p^{n-1}(p-1)$