Ramification and Decomposition - Applications

Ramification theory enables construction of extensions, determination of Galois groups, and solution of embedding problems.

TheoremKronecker-Weber Theorem

Every finite abelian extension of $\mathbb{Q}$ is contained in some cyclotomic field $\mathbb{Q}(\zeta_n)$ .

Proof sketch: Use ramification data. An abelian extension $K/\mathbb{Q}$ ramifies only at finitely many primes. The conductor $f_K$ (product of ramified primes with exponents) determines $K \subseteq \mathbb{Q}(\zeta_{f_K})$ .

This explicit class field theory for $\mathbb{Q}$ contrasts with general number fields, where Hilbert class fields provide abelian extensions.

ExampleSolvability by Radicals

Galois theory reduces solvability of polynomials to solvability of Galois groups. Ramification data constrains possible groups.

For quintic $f(x) = x^5 - x - 1$ : computing discriminant and ramification in the splitting field reveals $\text{Gal}(f) = S_5$ , proving insolvability by radicals.

Ramification patterns distinguish $A_5, S_5$ , and other transitive subgroups of $S_5$ .

TheoremClass Field Theory and Conductors

Every finite abelian extension $L/K$ has a conductor $\mathfrak{f}$ , an ideal of $K$ divisible by all ramified primes. The Artin map gives an isomorphism: $I_\mathfrak{f}/P_\mathfrak{f} \xrightarrow{\sim} \text{Gal}(L/K)$

where $I_\mathfrak{f}$ consists of ideals coprime to $\mathfrak{f}$ and $P_\mathfrak{f}$ is the subgroup of principal ideals satisfying local norm conditions.

This is the Artin reciprocity law, the main theorem of class field theory.

ExampleConstructing Class Fields

For $K = \mathbb{Q}(i)$ with class number 1, the Hilbert class field is $H_K = K$ (no extension needed).

For $K = \mathbb{Q}(\sqrt{-5})$ with class number 2: the Hilbert class field is $H_K = \mathbb{Q}(\sqrt{-5}, i)$ with $[H_K : K] = 2$ and $\text{Gal}(H_K/K) \cong \mathbb{Z}/2\mathbb{Z}$ .

All primes of $K$ split or remain prime in $H_K$ ; none ramify.

TheoremLocal-Global Principles

The Hasse-Minkowski theorem: a quadratic form over $\mathbb{Q}$ represents zero nontrivially if and only if it does so over $\mathbb{R}$ and all $\mathbb{Q}_p$ .

Ramification determines which completions require checking. For $ax^2 + by^2 + cz^2 = 0$ : check $p | 2abc$ and the real place.

This local-global principle fails for cubics (Selmer's $3x^3 + 4y^3 + 5z^3 = 0$ counterexample) but holds for many natural problems.

ExampleModular Forms and Ramification

The mod $p$ Galois representation attached to a modular form $f$ has conductor equal to the level of $f$ . Ramification of this representation encodes where $f$ is not an eigenform for Hecke operators.

This connection between ramification and modular forms is central to the proof of Fermat's Last Theorem via the modularity theorem.