User:Nuerk/Ramification theory

Ramification theory in Dedekind domains
Let $$A$$ be a Dedekind domain with quotient field $$F$$, let $$E/F$$ be a finite extension of $$F$$ and let $$B$$ be the integral closure of $$A$$ in $$E$$. Then $$B$$ is again a Dedekind domain and finitely generated as an $A$-module.

For each prime ideal $$\mathfrak{p} \neq 0$$ of $$A$$ the ideal $$\mathfrak{p}B$$ of $$B$$ is a product of prime ideals $$\mathfrak{p}B = \mathfrak{P}_1^{e_1}...\mathfrak{P}_r^{e_r}$$, and this decomposition is unique up to a reordering of the factors.

What is more, the prime ideals $$\mathfrak{P}_i$$ are precisely the prime ideals $$\mathfrak{P}$$ of $$B$$ that lie over $$\mathfrak{p}$$, i.e. satisfy $$\mathfrak{p} = \mathfrak{P} \cap A$$. In this case, we call $$\mathfrak{P}$$ a prime divisor of $$\mathfrak{p}$$, denoted by $$\mathfrak{P}|\mathfrak{p}$$.

The exponent $$e_i$$ is called the ramification index of $$\mathfrak{P}_i$$ over $$\mathfrak{p}$$ and the degree $$f_i = [B/ \mathfrak{P}_i : A/ \mathfrak{p}]$$ of the residue class field extension is called inertia degree of $$\mathfrak{P}_i$$ over $$\mathfrak{p}$$.

The prime ideal $$\mathfrak{p}$$ is said to split completely (or to be totally split) in $$E$$, if $$r = [E:F]$$ in the decomposition
 * $$\mathfrak{p}B = \prod_{i=1}^r \mathfrak{P}_i^{e_i}$$.

If $$r=1$$, then $$\mathfrak{p}$$ is called nonsplit.

The prime ideal $$\mathfrak{P}_i$$ is called unramified over $$A$$ (or over $$F$$) if $$e_i = 1$$ and if the residue class field extension $$^{\textstyle (B/\mathfrak{P}_i)}\big/_{\textstyle (A/\mathfrak{p})}$$ is separable. Otherwise it is called ramified, and totally ramified if furthermore $$f_i = 1$$.

The prime ideal $$\mathfrak{p}$$ is called unramified if all $$\mathfrak{P}_i$$ are unramified, else it is called ramified. The extension $$E/F$$ itself is called unramified if all prime ideals $$\mathfrak{p} \subset A$$ are unramified in $$E$$.

Separable field extensions
If the extension $$E/F$$ is furthermore separable then the fundamental identity
 * $$\sum_{i=1}^r e_i f_i = [E:F]$$

holds.

Let $$\theta \in B$$ be an integral, primitive element of the separable extension $$E/F$$, i.e. $$E = F[\theta]$$, and let $$p(X) \in A[X]$$ be the minimal polynomial of $$\theta$$. The conductor $$\mathfrak{c}$$ of $$A[\theta]$$ is defined by $$\mathfrak{c}=\{\alpha \in B : \alpha B \in A[\theta]\}$$. This is the largest ideal of $$B$$ that is contained in $$A[\theta]$$. It is always $$\mathfrak{c} \neq 0$$.

Let $$\mathfrak{p}$$ be a prime ideal of $$A$$ that is coprime to the conductor $$\mathfrak{c}$$ of $$A[\theta]$$. Then the prime ideals of $$B$$ that lie over $$\mathfrak{p}$$ can be explicitly constructed.

For this, let $$\bar{p}(X) = \prod_{i=1}^r \bar{p}_i(X)^{e_i}$$ be the decomposition of the polynomial $$\bar{p}(X) = p(X) \,\bmod\, \mathfrak{p}$$ in irreducible factors $$\bar{p}_i(X) = p_i(X) \,\bmod\, \mathfrak{p}$$, with $$p_i(X) \in A[X]$$ monic, over the residue field $$A/\mathfrak{p}$$. Then the prime ideals of $$B$$ that lie over $$\mathfrak{p}$$ are given by
 * $$\mathfrak{P}_i = \mathfrak{p}B + p_i(\theta)B, \quad i = 1,...,r$$.

The inertia degree $$f_i$$ of $$\mathfrak{P}_i$$ is equal to the degree of the polynomial $$\bar{p}_i(X)$$ and it is
 * $$\mathfrak{p}B = \mathfrak{P}_1^{e_1}...\mathfrak{P}_r^{e_r}$$.

It can be shown that, in the case where $$E/F$$ is separable, only finitely many prime ideals $$p$$ of $$A$$ ramify in $$E$$. The ramified ideals are described by the discriminant $$\mathfrak{d}$$ of $$B/A$$. This is the ideal of $$A$$ generated by the discriminants $$d(\omega_1,...,\omega_n)$$ of all bases $$\omega_1,...,\omega_n$$ of $$E/F$$ that are contained in $$B$$. The prime divisors of $$\mathfrak{d}$$ are precisely the prime ideals of $$A$$ that ramify in $$E$$.

Hilbert's ramification theory
Let in the following $$E/F$$ be a Galois extension with Galois group $$G = G(E/F)$$. Then $$G$$ acts on $$B$$, for $$\sigma a \in B$$ whenever $$a \in B$$ and $$\sigma \in G$$.

Let $$\sigma \in G$$. If $$\mathfrak{P} \subset B$$ is a prime ideal lying over $$\mathfrak{p} \subset A$$, then so is $$\sigma \mathfrak{P}$$, since
 * $$\sigma \mathfrak{P} \cap A = \sigma(\mathfrak{P} \cap A) = \sigma \mathfrak{p} = \mathfrak{p}$$.

The $$\sigma \mathfrak{P}$$, $$\sigma \in G$$, are called the prime ideals conjugate to $$\mathfrak{P}$$.

The Galois group acts transitively on the prime ideals $$\mathfrak{P}$$ of $$B$$ that lie over $$\mathfrak{p}$$, and so all these prime ideals are conjugate to each other.

Let $$\mathfrak{P}$$ be a prime ideal of $$B$$. The stabilizer
 * $$G_{\mathfrak{P}} = \{\sigma \in G : \sigma \mathfrak{P} = \mathfrak{P}\}$$

is called the decomposition group of $$\mathfrak{P}$$ over $$F$$. The fixed field
 * $$Z_\mathfrak{P} = \{x \in E : \sigma x = x \quad \forall \sigma \in G \}$$

is called the decomposition field of $$\mathfrak{P}$$ over $$F$$.

The number of different prime ideals over $$\mathfrak{p}$$ equals $$\left|G/G_\mathfrak{P}\right|$$. To see this, let $$\mathfrak{P}$$ be such a prime ideal and let $$\sigma$$ run through a transversal, i.e. a complete system of coset representatives, of $$G_\mathfrak{P}$$ in $$G$$. Then $$\sigma \mathfrak{P}$$ runs through the different prime ideals over $$\mathfrak{p}$$ exactly once, and so their number is indeed equal to $$\left|G/G_\mathfrak{P}\right|$$. From this, we also have the following equivalences:
 * $$G_\mathfrak{P} = 1 \iff Z_\mathfrak{P} = E \iff \mathfrak{p}$$ splits completely,
 * $$G_\mathfrak{P} = G \iff Z_\mathfrak{P} = F \iff \mathfrak{p}$$ is nonsplit.

The decomposition group of a prime ideal $$\sigma \mathfrak{P}$$ conjugated to $$\mathfrak{P}$$ is the conjugated subgroup
 * $$G_{\sigma \mathfrak{P}} = \sigma G_\mathfrak{P} \sigma^{-1}$$.

Furthermore, it follows from the transitivity of the Galois action that the inertia degrees $$f_1,...,f_r$$ and the ramification indices $$e_1,...,e_r$$ in the prime decomposition
 * $$\mathfrak{p} B = \mathfrak{P}_1^{e_1}...\mathfrak{P}_r^{e_r}$$

are independent of the index,
 * $$f_1 = ... = f_r = f$$, $$\quad e_1 = ... = e_r = e$$,

which simplifies the fundamental identity to
 * $$r e f = [E:F]$$.

The ramification index $$e$$ and the inertia degree $$f$$ admit a group-theoretic interpretation. Each $$\sigma \in G$$ induces, due to $$\sigma B = B$$ and $$\sigma \mathfrak{P} = \mathfrak{P}$$, an automorphism
 * $$\bar{\sigma} \colon\, B/\mathfrak{P} \to B/\mathfrak{P}, \; a\, \bmod\, \mathfrak{P} \mapsto \sigma a\, \bmod\, \mathfrak{P}$$

of the residue field $$B/\mathfrak{P}$$. If we set $$\kappa(\mathfrak{P}) = B/\mathfrak{P}$$ and $$\kappa(\mathfrak{p}) = A/\mathfrak{p}$$, then the extension $$\kappa(\mathfrak{P})/\kappa(\mathfrak{p})$$ is normal and there exists a surjective homomorphism
 * $$G_\mathfrak{P} \to G(\kappa(\mathfrak{P})/\kappa(\mathfrak{p}))$$.

The kernel $$I_\mathfrak{P} \subseteq G_\mathfrak{P}$$ of this homomorphism is called the inertia group of $$\mathfrak{P}$$ over $$F$$. Its fixed field
 * $$T_\mathfrak{P} = \{ x \in E : \sigma x = x \quad \forall \sigma \in I_\mathfrak{P}\}$$

is called the inertia field of $$\mathfrak{P}$$ over $$F$$.

We have the following chain of inclusions
 * $$F \subseteq Z_\mathfrak{P} \subseteq T_\mathfrak{P} \subseteq E$$

and the exact sequence
 * $$ 1 \to I_\mathfrak{P} \to G_\mathfrak{P} \to G(\kappa(\mathfrak{P})/\kappa(\mathfrak{P})) \to 1$$.

The extension $$T_\mathfrak{P}/Z_\mathfrak{P}$$ is normal and we have
 * $$G(T_\mathfrak{P}/Z_\mathfrak{P}) \cong G(\kappa(\mathfrak{P})/\kappa(\mathfrak{p}))$$, as well as
 * $$G(E/T_\mathfrak{P}) = I_\mathfrak{P}$$.

If the residue field extension $$\kappa(\mathfrak{P})/\kappa(\mathfrak{p})$$ is separable, then
 * $$\# I_\mathfrak{P} = [E:T_\mathfrak{P}] = e$$, and
 * $$(G_\mathfrak{P}:I_\mathfrak{P}) = [T_\mathfrak{P}:Z_\mathfrak{P}] = f$$.

Ramification theory in henselian fields
Let $$F$$ be a henselian field with respect to a non-Archimedean valuation $$v$$. If $$E/F$$ is a field extension, then there exists a unique extension $$w$$ of $$v$$ to a valuation of $$E$$. In the case that $$n = [L:K] < \infty$$, this extension is given by $$w(\alpha) = \frac{1}{n} v(N_{E/F}(\alpha))$$, where $$N$$ is the field norm.

Let $$A$$ (resp. $$B$$) be the valuation ring, $$\mathfrak{p}$$ (resp. $$\mathfrak{P}$$) its maximal ideal and let $$\kappa = A/\mathfrak{p}$$ (resp. $$\lambda = B/\mathfrak{P}$$) be the residual field of $$v$$ (resp. $$w$$). Then $$B$$ is the integral closure of $$A$$ in $$E$$ and we have the inclusions
 * $$v(F^\times) \subseteq w(E^\times)$$ and $$\kappa \subseteq \lambda$$.

The index $$e = e(w/v) = (w(E^\times) : v(F^\times))$$ is called the ramification index of the extension $$E/F$$ and the degree $$f = f(w/v) = [\lambda : \kappa]$$ its inertia degree.

It is always the case that $$[E:F] \geq e f$$. If, furthermore, $$v$$ is discrete, and the extension $$E/F$$ is separable, then we have equality, $$[E:F] = e f$$.

A finite extension $$E/F$$ is called unramified if the residue field extension $$\lambda / \kappa $$ is separable and if
 * $$[E:F]=[\lambda:\kappa]$$.

An arbitrary algebraic extension is called unramified if it can be written as the union of finite unramified subextensions.

Every subextension of an unramified extension is itself unramified. Likewise, the composite of two unramified extensions of $$F$$ is again unramified.

If $$E/F$$ is an algebraic extension, then the composite $$T$$ of all unramified subextensions of $$E/F$$ is called the maximal unramified subextension of $$E/F$$.

The residue field of $$T$$ is the separable hull $$\lambda_{s}$$ of $$\kappa$$ in the residue field extension $$\lambda / \kappa$$ of $$E/F$$, while the value group of $$T$$ is equal to that of $$F$$.

The maximal unramified extension $$F_{nr}/F$$ per se (nr = non ramifée) is the composite of all unramified extensions of $$F$$ in its algebraic closure $$\overline{F}$$. The residue field of $$F_{nr}$$ is the separable closure $$\overline{\kappa_s}$$ of $$\kappa$$.

Let in the following the residue field characteristic $$p = char(\kappa)$$ be positive.

An algebraic extension $$E/F$$ is called tamely ramified if the residue field extension $$\lambda / \kappa$$ is separable and if $$\operatorname{gcd}([E:T],p) = 1$$, which in the case where the extension is infinite is to mean that the degree of every finite subextension of $$E/T$$ is coprime to $$p$$.

A finite extension $$E/F$$ is tamely ramified if and only if $$E/T$$ is a radical extension, i.e. if there exist $$a_1,...,a_r \in E$$ and $$m_1,...,m_r \in \N$$, $$\operatorname{gcd}(m_i,p)=1$$, such that
 * $$E=T(\sqrt[m_1]{a_1},...,\sqrt[m_r]{a_r})$$.

In this setting the fundamental identity
 * $$[E:F] = e f$$

always holds.

It is every subextension of a tamely ramified extension again tamely ramified. Also, the composite of tamely ramified extensions is again tamely ramified.

If $$E/F$$ is an algebraic extension, then the composite $$V$$ of all tamely ramified subextensions of $$E/F$$ is called the maximal tamely ramified subextension of $$E/F$$.

If $$E/F$$ is finite and $$T=F$$, then the extension $$E/F$$ is called purely ramified. It is called wildly ramified if it is not tamely ramified, i.e. if $$V \neq E$$.

Ramification theory in general valued fields
Let $$F$$ be a field with valuation $$v$$.

Let $$v$$ be non-Archimedean and let $$E/F$$ be a finite extension. We denote an extension of $$v$$ to $$E$$ by $$w/v$$. Analogously to the henselian case, the ramification index of an extension $$w/v$$ is defined by
 * $$e_w = (w(E^\times) : v(F^\times))$$,

and its inertia degree by
 * $$f_w = [\lambda_w : \kappa]$$,

where $$\lambda_w$$ (resp. $$\kappa$$) denotes the residue field of $$w$$ (resp. $$v$$).

If $$v$$ is discrete and $$E/F$$ is separable, then the fundamental identity of valuation theory holds:
 * $$\sum_{w/v}e_w f_w = [E:F]$$.

Let in the following $$E/F$$ be a Galois extension with Galois group $$G=G(E/F)$$. Let $$v$$ be a valuation of $$F$$. Then $$G$$ acts on the set of extensions of $$v$$ to $$E$$, since, given such an extension $$w$$ of $$v$$ and a $$\sigma \in G$$, $$w \circ \sigma$$ also extends $$v$$. This action is transitive, i.e. any two extensions are conjugate.

The decomposition group of an extension $$w/v$$ to $$E$$ is defined by
 * $$G_w = G_w(E/F) = \{\sigma \in G(E/F) : w \circ \sigma = w\}$$.

If $$v$$ is a non-Archimedean valuation, then the decomposition group $$G_w$$ contains two more canonical subgroups, $$ G_w \supseteq I_w \supseteq R_w$$, defined as follows. Let $$B_w$$ be the valuation ring of $$w$$ and $$\mathfrak{P}_w$$ its maximal ideal. Then the inertia group of $$w/v$$ is defined by
 * $$I_w = I_w(E/F) = \{\sigma \in G_w : \sigma x \equiv x \, \bmod \, \mathfrak{P}_w \quad \forall x \in B_w\}$$

and the ramification group of $$w/v$$ by
 * $$R_w = R_w(E/F) = \{\sigma \in G_w : \frac{\sigma x}{x} \equiv 1 \, \bmod \, \mathfrak{P}_w \quad \forall x \in E^\times \}$$.

The fixed field of $$G_w$$,
 * $$Z_w = Z_w(E/F) = \{x \in E : \sigma x = x \quad \forall \sigma \in G_w \}$$,

is called the decomposition field of $$w$$ over $$F$$. The fixed field of $$I_w$$,
 * $$T_w = T_w(E/F) = \{x \in E : \sigma x = x \quad \forall \sigma \in I_w \}$$,

is called the inertia field of $$w$$ over $$F$$. And the fixed field of $$R_w$$,
 * $$V_w = V_w(E/F) = \{x \in E : \sigma x = x \quad \forall \sigma \in R_w \}$$,

is called the ramification field of $$w$$ over $$F$$.

It is $$T_w/Z_w$$ the maximal unramified subextension of $$E/Z_w$$, and $$V_w/Z_w$$ is the maximal tamely ramified subextension of $$E/Z_w$$.

Higher ramification groups
Let $$E/F$$ be a finite Galois extension with Galois group $$G$$. Let $$v$$ be a discrete normalized valuation of $$F$$ with positive residue field characteristic $$p$$, such that there is a unique extension $$w$$ of $$v$$ to $$E$$. Let $$v_E$$ denote the corresponding normalized valuation of $$E$$, and $$B$$ its valuation ring.

Then for every real number $$s \geq -1$$ the $$s$$-th ramification group of $$E/F$$ is defined by
 * $$G_s = G_s(E/F) = \{\sigma \in G : v_E(\sigma a - a) \geq s+1 \quad \forall a \in B\}$$.

It follows that $$G_{-1} = G$$, $$G_0$$ is the inertia group $$I = I(E/F)$$, and $$G_1$$ is the ramification group $$R = R(E/F)$$.

The ramification groups form a chain
 * $$G = G_{-1} \supseteq G_0 \supseteq G_1 \supseteq G_2 \supseteq ...$$

of normal subgroups of $$G$$.

One can prove the following theorem for the factor groups $$G_s/G_{s+1}$$. Let $$\pi_E \in B$$ be a prime element of $$E$$. Then for every integer $$s \geq 0$$ the mapping
 * $$G_s/G_{s+1} \to U_E^{(s)}/U_E^{(s+1)}, \; \sigma \mapsto \frac{\sigma \pi_E}{\pi_E}$$

is an injective homomorphism. This homomorphism is independent of the choice of the prime element $$\pi_E$$. Here, $$U_E^{(s)}$$ denotes the $$s$$-th higher unit group of $$E$$, i.e. $$U_E^{(0)} = B^\times$$ and $$U_E^{(s)} = 1 + \pi_E^s B$$ for $$s \geq 1$$.