Mathematics Lecture Notes

DefinitionInertia Group

For a Galois extension $L/K$ and prime $\mathfrak{P}$ of $L$ above $\mathfrak{p}$ of $K$ , the inertia group is: $I_\mathfrak{P} = \{\sigma \in D_\mathfrak{P} : \sigma(\alpha) \equiv \alpha \pmod{\mathfrak{P}} \text{ for all } \alpha \in \mathcal{O}_L\}$

This is a normal subgroup of $D_\mathfrak{P}$ . The extension is unramified at $\mathfrak{p}$ if and only if $I_\mathfrak{P} = \{1\}$ .

The quotient $D_\mathfrak{P}/I_\mathfrak{P} \cong \text{Gal}((\mathcal{O}_L/\mathfrak{P})/(\mathcal{O}_K/\mathfrak{p}))$ is cyclic, generated by the Frobenius element.

TheoremRamification Formula

In a Galois extension $L/K$ , all primes $\mathfrak{P}_i$ above $\mathfrak{p}$ have the same ramification index $e$ and inertia degree $f$ . Moreover: $efg = n = [L : K]$

where $g$ is the number of primes above $\mathfrak{p}$ . The orders satisfy: $|D_\mathfrak{P}| = ef, \quad |I_\mathfrak{P}| = e$

ExampleCyclotomic Extensions

For $L = \mathbb{Q}(\zeta_p)$ with $p$ prime, the only ramified prime is $(p)$ , which factors as: $(p) = (1 - \zeta_p)^{p-1}$

with ramification index $e = p - 1$ , inertia degree $f = 1$ , and $g = 1$ . All other rational primes are unramified.

For a prime $q \neq p$ , the decomposition depends on the order of $q$ modulo $p$ (smallest $f$ with $q^f \equiv 1 \pmod{p}$ ).

DefinitionDifferent and Discriminant

The different $\mathfrak{D}_{L/K}$ is the inverse of the ideal: $\{\alpha \in L : \text{Tr}_{L/K}(\alpha \mathcal{O}_L) \subseteq \mathcal{O}_K\}$

The discriminant is $\Delta_{L/K} = N_{L/K}(\mathfrak{D}_{L/K})$ . These measure ramification: $\mathfrak{D}_{L/K} = \prod_{\mathfrak{P}} \mathfrak{P}^{e_\mathfrak{P} - 1 + \delta_\mathfrak{P}}$

where $\delta_\mathfrak{P} \geq 0$ measures wild ramification ( $\delta_\mathfrak{P} > 0$ when $e_\mathfrak{P}$ is divisible by the residue characteristic).

Remark

Tame vs. wild ramification: If $e_\mathfrak{P}$ is coprime to the residue characteristic, ramification is tame and $\delta_\mathfrak{P} = 0$ . Otherwise, it's wild with $\delta_\mathfrak{P} > 0$ .

Wild ramification is more complicated, involving higher ramification groups. Tame ramification is well-understood via Kummer theory.

TheoremHilbert's Ramification Theory

In an abelian extension $L/K$ , the prime $\mathfrak{p}$ splits completely if and only if $\mathfrak{p}$ is a norm from $L$ : there exists $\beta \in L$ with $N_{L/K}(\beta) \in \mathfrak{p}$ .

This is a special case of the Artin reciprocity law, connecting splitting behavior to ideal-theoretic conditions.

ExampleRamification in Number Field Towers

Consider $\mathbb{Q} \subset K \subset L$ . If $\mathfrak{p}$ ramifies in $L/\mathbb{Q}$ , it may ramify in $K/\mathbb{Q}$ , in $L/K$ , or both.

The ramification indices satisfy: $e_{L/\mathbb{Q}} = e_{K/\mathbb{Q}} \cdot e_{L/K}$ .

For $\mathbb{Q} \subset \mathbb{Q}(i) \subset \mathbb{Q}(i, \sqrt{2+i})$ : the prime $(2)$ ramifies at each level, with ramification indices multiplying appropriately.

Math Notes

Ramification and Decomposition - Key Properties