Ramification, decomposition and inertia groups, the discriminant and the different

Intuition Beginner

When you enlarge the number system, every ordinary prime meets one of three fates. It can stay prime (it is then called inert), it can shatter into several new prime pieces (it splits), or it can ramify — collapsing into a repeated power of a single new prime, like a root where a curve just grazes the axis.

Ramification is the distinctive fate. In the Gaussian integers $\mathbb{Z}[i]$, the ordinary prime $2$ becomes $(1+i{)}^2$, a perfect square of the new prime $1+i$. The exponent $2$ is the signal: the prime has ramified. By contrast $3$ stays whole and $5$ splits cleanly as $(2+i)(2-i)$.

A single integer called the discriminant detects ramification in advance. The discriminant of $\mathbb{Q}(i)$ is $-4$, and its only prime divisor is $2$ — exactly the ramified prime. The discriminant rings a bell at the ramified primes and is silent at the others, so ramification alone among the three fates can be read off before any factorisation is computed.

Visual Beginner

The three fates of an ordinary prime inside an extended number system. Only the ramified column carries a repeated exponent, and only that prime divides the discriminant.

Worked example Beginner

Work in the Gaussian integers $\mathbb{Z}[i]$, the numbers $a+bi$ with $a,b$ whole and $i^2=-1$. The fate of each ordinary prime is read off by factoring $x^2+1$ modulo that prime.

Prime $2$. Compute $(1+i)(1-i)=1-i^2=1-(-1)=2$. The two factors differ by a unit, since $1-i=-i(1+i)$ and $-i$ is a unit. So $2=-i(1+i{)}^2$, a unit times a perfect square. The prime $2$ has ramified, with exponent $2$.

Prime $3$. Reduce $x^2+1$ modulo $3$: the squares modulo $3$ are $0$ and $1$, never $2\equiv -1$. So $x^2+1$ has no root modulo $3$, and $3$ stays prime inside $\mathbb{Z}[i]$. It is inert.

Prime $5$. Here $2^2+1=5$, giving the factorisation $5=(2+i)(2-i)$ into two genuinely distinct Gaussian primes. The prime $5$ has split.

What this tells us: of $2,3,5$, only $2$ ramifies. The discriminant of $\mathbb{Q}(i)$ is $-4$, whose sole prime divisor is $2$. The discriminant pointed precisely at the ramified prime and was silent at the inert prime $3$ and the split prime $5$.

Check your understanding Beginner

Formal definition Intermediate+

Let $L/K$ be a finite separable extension of number fields with rings of integers ${\mathcal{O}}_K\subseteq {\mathcal{O}}_L$, and let $\mathfrak{p}$ be a nonzero prime ideal of ${\mathcal{O}}_K$. Unique factorisation of ideals 21.17.01 gives

$$\mathfrak{p}{\mathcal{O}}_L={\mathfrak{P}}_1^{e_1}{\mathfrak{P}}_2^{e_2}\cdots {\mathfrak{P}}_g^{e_g},e_i\ge 1,$$

where the ${\mathfrak{P}}_i$ are the distinct prime ideals of ${\mathcal{O}}_L$ lying over $\mathfrak{p}$ ^{[Neukirch Ch. I §8]}. The integer $e_i=e({\mathfrak{P}}_i/\mathfrak{p})$ is the ramification index, and

$$f_i=f({\mathfrak{P}}_i/\mathfrak{p})=[\kappa ({\mathfrak{P}}_i):\kappa (\mathfrak{p})],\kappa ({\mathfrak{P}}_i):={\mathcal{O}}_L/{\mathfrak{P}}_i,\kappa (\mathfrak{p}):={\mathcal{O}}_K/\mathfrak{p},$$

is the residue degree (or inertia degree). The residue fields $\kappa ({\mathfrak{P}}_i),\kappa (\mathfrak{p})$ are finite fields, so each extension $\kappa ({\mathfrak{P}}_i)/\kappa (\mathfrak{p})$ is Galois and cyclic.

The prime $\mathfrak{p}$ is ramified in $L$ if some $e_i\ge 2$, and unramified otherwise. The three archetypal behaviours when $L/K$ is Galois of degree $n$ (so all $e_i=e$ and all $f_i=f$ are equal) are: totally ramified ($g=1,f=1,e=n$); inert ($g=1,e=1,f=n$); totally split ($e=f=1,g=n$). Taking the base $K=\mathbb{Q}$ recovers the fate of each rational prime.

Assume now that $L/K$ is Galois with group $G=\mathrm{Gal}(L/K)$. The group $G$ acts on the set $\{{\mathfrak{P}}_1,\dots ,{\mathfrak{P}}_g\}$ of primes above $\mathfrak{p}$, and this action is transitive ^{[Neukirch Ch. II]}. For a chosen prime $\mathfrak{P}={\mathfrak{P}}_1$ above $\mathfrak{p}$, the decomposition group is the stabiliser

$$D_{\mathfrak{P}}=\{\sigma \in G:\sigma (\mathfrak{P})=\mathfrak{P}\},$$

of order $|D_{\mathfrak{P}}|=ef$, with orbit size $[G:D_{\mathfrak{P}}]=g$. Every $\sigma \in D_{\mathfrak{P}}$ preserves $\mathfrak{P}$, hence descends to an automorphism $\bar{\sigma }$ of the residue field $\kappa (\mathfrak{P})$ fixing $\kappa (\mathfrak{p})$ pointwise. This yields a homomorphism

$$D_{\mathfrak{P}}⟶\mathrm{Gal}(\kappa (\mathfrak{P})/\kappa (\mathfrak{p})),\sigma \mapsto \bar{\sigma },$$

whose kernel is the inertia group

$$I_{\mathfrak{P}}=\{\sigma \in D_{\mathfrak{P}}:\sigma (x)\equiv x(\mathrm{mod}\mathfrak{P})\text{ for all }x\in {\mathcal{O}}_L\}.$$

Because the residue extension is an extension of finite fields, hence separable, this map is surjective, giving the fundamental exact sequence

$$1⟶I_{\mathfrak{P}}⟶D_{\mathfrak{P}}⟶\mathrm{Gal}(\kappa (\mathfrak{P})/\kappa (\mathfrak{p}))⟶1,$$

with $|I_{\mathfrak{P}}|=e$ and $[D_{\mathfrak{P}}:I_{\mathfrak{P}}]=f$ ^{[Serre Ch. I; Neukirch Ch. II]}.

The (absolute) discriminant of a number field $K$ of degree $n$, with integral basis ${\omega }_1,\dots ,{\omega }_n$ and embeddings ${\sigma }_1,\dots ,{\sigma }_n$, is the nonzero integer

$$d_K=\det ({\sigma }_i({\omega }_j){)}^2=\det ({\mathrm{Tr}}_{K/\mathbb{Q}}({\omega }_i{\omega }_j))$$

introduced in 21.17.01. The different of the extension $L/K$ is the ideal ${\mathfrak{D}}_{L/K}\subseteq {\mathcal{O}}_L$ inverse to the codifferent

$${\mathfrak{D}}_{L/K}^{-1}=\{x\in L:{\mathrm{Tr}}_{L/K}(x{\mathcal{O}}_L)\subseteq {\mathcal{O}}_K\}.$$

The relative discriminant is the ideal ${\mathfrak{d}}_{L/K}=N_{L/K}({\mathfrak{D}}_{L/K})\subseteq {\mathcal{O}}_K$, the norm of the different. The discriminant-different formula relates the absolute discriminants:

$$d_L=d_K^{[L:K]}\cdot N_{K/\mathbb{Q}}({\mathfrak{d}}_{L/K})$$

^{[Neukirch Ch. III §3; Serre Ch. IV]}. For $K=\mathbb{Q}$ this reduces to $|d_L|=N_{L/\mathbb{Q}}({\mathfrak{D}}_{L/\mathbb{Q}})$: the absolute value of the discriminant is the norm of the different.

Counterexamples to common slips

• The decomposition group depends on the choice of $\mathfrak{P}$ above $\mathfrak{p}$, but only up to conjugation: choosing ${\mathfrak{P}}^{\prime }=\tau (\mathfrak{P})$ gives $D_{{\mathfrak{P}}^{\prime }}=\tau D_{\mathfrak{P}}{\tau }^{-1}$. The inertia group behaves the same way. For abelian $G$ all choices give the same subgroup.
• Surjectivity of $D_{\mathfrak{P}}\to \mathrm{Gal}(\kappa (\mathfrak{P})/\kappa (\mathfrak{p}))$ can fail for extensions of Dedekind domains whose residue extension is inseparable. Over number fields residue fields are finite, hence perfect, and surjectivity always holds.
• A prime $\mathfrak{p}$ may be unramified yet not split, and may be ramified yet not totally ramified. Inertness ($g=1,e=1$) and total ramification ($g=1,f=1$) are opposite extremes; the generic unramified prime is split.

Key theorem with proof Intermediate+

Theorem (Dedekind's factorisation theorem). Let $K=\mathbb{Q}(\alpha )$ with $\alpha$ an algebraic integer and minimal polynomial $f(x)\in \mathbb{Z}[x]$, and let $p$ be a rational prime with $p\nmid [{\mathcal{O}}_K:\mathbb{Z}[\alpha ]]$. Factor the reduction $\bar{f}\in {\mathbb{F}}_p[x]$ as

$$\bar{f}(x)={\bar{g}}_1(x{)}^{e_1}{\bar{g}}_2(x{)}^{e_2}\cdots {\bar{g}}_r(x{)}^{e_r},$$

with the ${\bar{g}}_i$ distinct monic irreducibles, and lift each ${\bar{g}}_i$ to a monic $g_i\in \mathbb{Z}[x]$. Then

$$p{\mathcal{O}}_K={\mathfrak{P}}_1^{e_1}{\mathfrak{P}}_2^{e_2}\cdots {\mathfrak{P}}_r^{e_r},{\mathfrak{P}}_i=(p,g_i(\alpha )),f({\mathfrak{P}}_i/p)=\mathrm{deg}{\bar{g}}_i.$$

In particular $p$ ramifies in ${\mathcal{O}}_K$ if and only if $\bar{f}$ has a repeated irreducible factor, if and only if $\gcd (\bar{f},\bar{f}{}^{\prime })\ne 1$ in ${\mathbb{F}}_p[x]$ ^{[Marcus Ch. 3; Neukirch Ch. I §8]}.

Proof. The evaluation map $\mathbb{Z}[x]\to \mathbb{Z}[\alpha ]$, $h\mapsto h(\alpha )$, has kernel $(f)$, so $\mathbb{Z}[\alpha ]\cong \mathbb{Z}[x]/(f)$. Reduce modulo $p$. Because $p$ does not divide the index $[{\mathcal{O}}_K:\mathbb{Z}[\alpha ]]$, the finite abelian group ${\mathcal{O}}_K/\mathbb{Z}[\alpha ]$ has order prime to $p$, and tensoring with ${\mathbb{F}}_p$ annihilates it. Hence reduction modulo $p$ identifies

$${\mathcal{O}}_K/p{\mathcal{O}}_K\cong \mathbb{Z}[\alpha ]/p\mathbb{Z}[\alpha ]\cong {\mathbb{F}}_p[x]/(\bar{f}).$$

Factor $\bar{f}={\bar{g}}_1^{e_1}\cdots {\bar{g}}_r^{e_r}$ in ${\mathbb{F}}_p[x]$. The Chinese remainder theorem splits the quotient as a product of local Artinian rings,

$${\mathbb{F}}_p[x]/(\bar{f})\cong \prod _{i=1}^r{\mathbb{F}}_p[x]/({\bar{g}}_i^{e_i}).$$

Each factor ${\mathbb{F}}_p[x]/({\bar{g}}_i^{e_i})$ is local with maximal ideal generated by ${\bar{g}}_i$ and residue field ${\mathbb{F}}_p[x]/({\bar{g}}_i)$, a finite field of $p^{\mathrm{deg}{\bar{g}}_i}$ elements. Pulling these factors back through the evaluation map $x\mapsto \alpha$ gives the prime ideals ${\mathfrak{P}}_i=(p,g_i(\alpha ))$ of ${\mathcal{O}}_K$ above $p$, with

$$\kappa ({\mathfrak{P}}_i)={\mathcal{O}}_K/{\mathfrak{P}}_i\cong {\mathbb{F}}_p[x]/({\bar{g}}_i),$$

so the residue degree is $f({\mathfrak{P}}_i/p)=\mathrm{deg}{\bar{g}}_i$, and the exponent $e_i$ is read off from the power of ${\bar{g}}_i$ in $\bar{f}$. This is exactly the prime factorisation of $p{\mathcal{O}}_K$.

For the ramification criterion, $p$ ramifies precisely when some $e_i\ge 2$, which holds precisely when $\bar{f}$ has a repeated irreducible factor. A polynomial over a field has a repeated factor if and only if it shares a root with its formal derivative, that is if and only if $\gcd (\bar{f},\bar{f}{}^{\prime })\ne 1$. $\square$

Corollary (the discriminant detects ramification). With $K=\mathbb{Q}(\alpha )$ as above and $p\nmid [{\mathcal{O}}_K:\mathbb{Z}[\alpha ]]$, the prime $p$ ramifies if and only if $p\mid \mathrm{disc}(f)$. Dropping the index condition, a prime $p$ ramifies in ${\mathcal{O}}_K$ if and only if $p\mid d_K$ (Dedekind discriminant theorem, proved in the Full proof set).

Bridge. Dedekind's factorisation theorem builds toward the local-field theory of 21.02.07, where the factorisation of $p{\mathcal{O}}_K$ is read off one completion at a time, and appears again in the étale fundamental group of schemes 04.02.01, where the decomposition group of a prime becomes the Galois group of a complete discretely valued field. The foundational reason a single integer detects all of ramification is that the discriminant is the determinant of the trace pairing, and a prime divides this determinant exactly when the pairing degenerates modulo $p$ — this is exactly the repeated-factor condition on $\bar{f}$. The central insight is that the three fates (split, inert, ramified) reduce to the factorisation type of one polynomial modulo $p$, and putting these together with the Galois action yields the decomposition and inertia groups as the mechanism governing that factorisation. The bridge is that the discriminant, the different, and the ramification indices are three readings of a single degeneracy of the trace form.

Exercises Intermediate+

Lean formalization Intermediate+

lean_status: none is recorded because the objects central to this unit — the different ideal ${\mathfrak{D}}_{L/K}$ defined as the inverse of the trace-dual codifferent, the decomposition and inertia subgroups of the Galois group with the exact sequence $1\to I_{\mathfrak{P}}\to D_{\mathfrak{P}}\to \mathrm{Gal}(\kappa (\mathfrak{P})/\kappa (\mathfrak{p}))\to 1$, Hilbert's formula for the tame different, and the discriminant-different formula $d_L=d_K^{[L:K]}\cdot N_{K/\mathbb{Q}}({\mathfrak{d}}_{L/K})$ — are not assembled as a single formal development in Mathlib, although the field discriminant (NumberField.discr), the Dedekind-domain infrastructure, and isolated pieces of prime ideal factorisation are present. The gap is upstream contribution work, described in the unit metadata; until it closes the unit is human-reviewed against the Serre and Neukirch sources.

Advanced results Master

Theorem (Dedekind discriminant theorem). A rational prime $p$ ramifies in ${\mathcal{O}}_K$ if and only if $p\mid d_K$. In particular only finitely many primes ramify, and they are exactly the prime divisors of the discriminant. Combined with Hermite-Minkowski finiteness (only finitely many fields of bounded discriminant) this makes the discriminant the coarse but faithful invariant controlling ramification ^{[Neukirch Ch. I §8]}. The proof, via degeneracy of the trace pairing modulo $p$, is given in the Full proof set.

Theorem (tame and wild ramification; Hilbert's formula for the different). Let $\mathfrak{P}$ lie over $\mathfrak{p}$ with ramification index $e$. The exponent of $\mathfrak{P}$ in the different ${\mathfrak{D}}_{L/K}$ is at least $e-1$, with equality when $\mathfrak{P}$ is tamely ramified (that is, when the residue characteristic does not divide $e$). A prime divides the different if and only if it is ramified, so the different and the discriminant detect exactly the same set of primes — the different refining the discriminant by recording, in the extension $L/K$ itself rather than over $\mathbb{Q}$, the precise local contribution of each ramified prime ^{[Serre Ch. VI; Neukirch Ch. III]}. In wild ramification the excess over $e-1$ is governed by the ramification groups $G\supseteq G_0=I_{\mathfrak{P}}\supseteq G_1\supseteq G_2\supseteq \cdots$, a descending filtration of $D_{\mathfrak{P}}$ measured in the lower numbering, and Hilbert's formula reads

$$v_{\mathfrak{P}}({\mathfrak{D}}_{L/K})=\sum _{i\ge 0}(|G_i|-1).$$

The Hasse-Arf theorem forces the jumps of this filtration in the upper numbering to be integers for abelian extensions — the bridge to local class field theory 21.02.07.

Theorem (Frobenius and the decomposition law). For an unramified prime $\mathfrak{p}$ ($e=1$, so $I_{\mathfrak{P}}=1$), the exact sequence identifies $D_{\mathfrak{P}}$ with $\mathrm{Gal}(\kappa (\mathfrak{P})/\kappa (\mathfrak{p}))$, and the preimage of the Frobenius $x\mapsto x^{N\mathfrak{p}}$ is a single element ${\mathrm{Frob}}_{\mathfrak{P}}\in G$. Its conjugacy class depends only on $\mathfrak{p}$, not on $\mathfrak{P}$. The Frobenius-distribution theorem of Frobenius, sharpened by Chebotarev to the full density statement, asserts that every conjugacy class of $G$ arises as the Frobenius class of a positive-density set of primes — the engine of all arithmetic statistics and the gateway to 21.03.03 ^{[Frobenius 1896]}.

Synthesis. Ramification theory rests on the single observation that the factorisation of $\mathfrak{p}{\mathcal{O}}_L$ is encoded in the degeneracy of the trace form modulo $\mathfrak{p}$, and the foundational reason this works is that the discriminant, the different, and the ramification indices are three readings of that one degeneracy — the discriminant over $\mathbb{Q}$, the different over $L/K$, the indices prime by prime. This is exactly the content of the discriminant-different formula $d_L=d_K^{[L:K]}\cdot N_{K/\mathbb{Q}}({\mathfrak{d}}_{L/K})$. The central insight is that the Galois group acts transitively on the primes above $\mathfrak{p}$, and the decomposition and inertia groups are the stabilisers of this action, so ramification generalises to an exact sequence $1\to I_{\mathfrak{P}}\to D_{\mathfrak{P}}\to \mathrm{Gal}(\kappa (\mathfrak{P})/\kappa (\mathfrak{p}))\to 1$ that splits the extension into a ramified part, an unramified part, and a residue-field part. The bridge is that this sequence builds toward the local theory of 21.02.07 — where $D_{\mathfrak{P}}=\mathrm{Gal}(L_{\mathfrak{P}}/K_{\mathfrak{p}})$ literally — and appears again in the étale fundamental group of 04.02.01, where it is dual to the splitting of a geometric cover; putting these together, the tame-wild dichotomy and the Hasse-Arf theorem promote the whole picture to the engine of class field theory and of the analytic distribution of primes in 21.03.03.

Full proof set Master

Proposition 1 (the decomposition–inertia exact sequence). Let $L/K$ be a Galois extension of number fields with group $G$, and let $\mathfrak{P}$ lie over $\mathfrak{p}$. The residue map $\phi :D_{\mathfrak{P}}\to \mathrm{Gal}(\kappa (\mathfrak{P})/\kappa (\mathfrak{p}))$, $\sigma \mapsto \bar{\sigma }$, is a surjective homomorphism with kernel $I_{\mathfrak{P}}$, giving the exact sequence

$$1\to I_{\mathfrak{P}}\to D_{\mathfrak{P}}\to \mathrm{Gal}(\kappa (\mathfrak{P})/\kappa (\mathfrak{p}))\to 1.$$

Proof. For $\sigma \in D_{\mathfrak{P}}$, the equality $\sigma (\mathfrak{P})=\mathfrak{P}$ lets $\sigma$ descend to a map $\bar{\sigma }(x\mathrm{mod}\mathfrak{P})=\sigma (x)\mathrm{mod}\mathfrak{P}$ of ${\mathcal{O}}_L/\mathfrak{P}=\kappa (\mathfrak{P})$. Because $\sigma$ fixes $K$ it fixes the subfield $\kappa (\mathfrak{p})$, and $\bar{\sigma }$ is an automorphism since $\sigma$ is. The identity $\overline{\sigma \tau }=\bar{\sigma }\bar{\tau }$ makes $\phi$ a homomorphism, and its kernel is $\{\sigma :\sigma (x)\equiv x(\mathrm{mod}\mathfrak{P})\text{ for all }x\in {\mathcal{O}}_L\}=I_{\mathfrak{P}}$.

For surjectivity, pass to the completions. The decomposition group is canonically $D_{\mathfrak{P}}\cong \mathrm{Gal}(L_{\mathfrak{P}}/K_{\mathfrak{p}})$, where $L_{\mathfrak{P}},K_{\mathfrak{p}}$ are complete discretely valued fields with the same residue fields $\kappa (\mathfrak{P}),\kappa (\mathfrak{p})$. The residue extension $\kappa (\mathfrak{P})/\kappa (\mathfrak{p})$ is an extension of finite fields, hence cyclic with generator the Frobenius $x\mapsto x^{N\mathfrak{p}}$. Let $\bar{\alpha }$ generate $\kappa (\mathfrak{P})$ over $\kappa (\mathfrak{p})$ with separable minimal polynomial $\bar{h}\in \kappa (\mathfrak{p})[t]$, and lift $\bar{h}$ to a monic $h\in {\mathcal{O}}_{K_{\mathfrak{p}}}[t]$ and $\bar{\alpha }$ to $\alpha \in {\mathcal{O}}_{L_{\mathfrak{P}}}$ by surjectivity of the reduction ${\mathcal{O}}_{L_{\mathfrak{P}}}\to \kappa (\mathfrak{P})$. Hensel's lemma lifts the factorisation of $\bar{h}$ to a root ${\alpha }^{\prime }$ of $h$ in ${\mathcal{O}}_{L_{\mathfrak{P}}}$ reducing to $\bar{\alpha }$, and any automorphism of $\kappa (\mathfrak{P})$ over $\kappa (\mathfrak{p})$ (sending $\bar{\alpha }\mapsto \bar{\beta }$, another root of $\bar{h}$) lifts, again by Hensel, to an automorphism of $L_{\mathfrak{P}}$ over $K_{\mathfrak{p}}$ sending ${\alpha }^{\prime }\mapsto {\beta }^{\prime }$. Hence $\phi$ is onto. $\square$

Proposition 2 (Hilbert's formula for the tame different). If $\mathfrak{P}$ lies tamely ramified over $\mathfrak{p}$ with index $e$, then the exponent of $\mathfrak{P}$ in ${\mathfrak{D}}_{L/K}$ equals $e-1$.

Proof. Localise and complete at $\mathfrak{P}$. The complete discrete valuation ring ${\mathcal{O}}_{L_{\mathfrak{P}}}$ has uniformiser ${\pi }_L$ and $\mathfrak{p}{\mathcal{O}}_{L_{\mathfrak{P}}}=({\pi }_L^e)$. Let $L_{\mathfrak{P}}^{\mathrm{ur}}/K_{\mathfrak{p}}$ be the maximal unramified subextension, so that ${\pi }_L$ is totally ramified over $L_{\mathfrak{P}}^{\mathrm{ur}}$ of degree $e$. Because the ramification is tame, the residue characteristic does not divide $e$, and ${\pi }_L$ is a root of an Eisenstein polynomial $F(t)=t^e-u{\pi }_K$ (after adjoining a uniformiser ${\pi }_K$ of $K_{\mathfrak{p}}$ and a unit $u$) whose derivative is $F^{\prime }(t)=e{t}^{e-1}$. Evaluating at $t={\pi }_L$ gives $v_{\mathfrak{P}}(F^{\prime }({\pi }_L))=v_{\mathfrak{P}}(e)+(e-1)v_{\mathfrak{P}}({\pi }_L)=0+(e-1)$, since tameness gives $v_{\mathfrak{P}}(e)=0$. The standard identity $v_{\mathfrak{P}}({\mathfrak{D}}_{L_{\mathfrak{P}}/L_{\mathfrak{P}}^{\mathrm{ur}}})=v_{\mathfrak{P}}(F^{\prime }({\pi }_L))$ (the different is generated by $F^{\prime }({\pi }_L)$ for an Eisenstein generator) yields the exponent $e-1$ for the totally ramified part; the unramified part contributes $0$. Hence $v_{\mathfrak{P}}({\mathfrak{D}}_{L/K})=e-1$. $\square$

Proposition 3 (Dedekind discriminant theorem: $p$ ramifies iff $p\mid d_K$). For a number field $K$ and rational prime $p$, the prime $p$ ramifies in ${\mathcal{O}}_K$ if and only if $p$ divides $d_K$.

Proof. Let ${\omega }_1,\dots ,{\omega }_n$ be an integral basis. The Gram matrix $({\mathrm{Tr}}_{K/\mathbb{Q}}({\omega }_i{\omega }_j))$ has determinant $d_K$. Reduce modulo $p$: this is the Gram matrix of the trace pairing on the finite ${\mathbb{F}}_p$-algebra ${\bar{\mathcal{O}}}_K:={\mathcal{O}}_K{\otimes }_{\mathbb{Z}}{\mathbb{F}}_p$, and $p\mid d_K$ if and only if this reduced pairing is degenerate.

Factor $p{\mathcal{O}}_K={\prod }_i{\mathfrak{P}}_i^{e_i}$. The Chinese remainder theorem gives ${\bar{\mathcal{O}}}_K\cong {\prod }_i{\mathcal{O}}_K/{\mathfrak{P}}_i^{e_i}$ as ${\mathbb{F}}_p$-algebras. A finite ${\mathbb{F}}_p$-algebra is a product of separable field extensions of ${\mathbb{F}}_p$ if and only if it is reduced, if and only if every $e_i=1$. For an Artin algebra over a perfect field, the trace pairing is nondegenerate on a factor ${\mathcal{O}}_K/{\mathfrak{P}}_i^{e_i}$ if and only if that factor is a field, that is $e_i=1$: when $e_i\ge 2$ the local factor has nilpotents and the trace pairing identifies as degenerate (the trace of a nilpotent against the unit vanishes). Therefore the reduced pairing is degenerate if and only if some $e_i\ge 2$, that is if and only if $p$ ramifies. Hence $p\mid d_K⟺p$ ramifies. $\square$

Connections Master

• Algebraic number theory foundations 21.17.01. This unit takes the Dedekind domain ${\mathcal{O}}_K$, the unique factorisation of ideals, the discriminant $d_K$, and the norm $N(\mathfrak{a})=|{\mathcal{O}}_K/\mathfrak{a}|$ from 21.17.01 as its raw material and asks what becomes of a single prime ideal under a field extension. The fundamental identity $\sum e_i{f}_i=n$ and the discriminant-different formula are the first refinement of that foundation, translating the ideal-theoretic structure of 21.17.01 into ramification data.

• Fields and Galois theory 01.04.01. The decomposition group $D_{\mathfrak{P}}$ and inertia group $I_{\mathfrak{P}}$ are subgroups of the Galois group $G=\mathrm{Gal}(L/K)$ of 01.04.01, and the exact sequence $1\to I_{\mathfrak{P}}\to D_{\mathfrak{P}}\to \mathrm{Gal}(\kappa (\mathfrak{P})/\kappa (\mathfrak{p}))\to 1$ refines the group-theoretic structure of 01.04.01 by measuring how $G$ sits over a single prime. The Frobenius element is a canonical generator of the cyclic Galois group of a finite-field extension.

• Schemes and the étale fundamental group 04.02.01. The factorisation of $\mathfrak{p}{\mathcal{O}}_L$ is the arithmetic shadow of how the map $\mathrm{Spec}{\mathcal{O}}_L\to \mathrm{Spec}{\mathcal{O}}_K$ of 04.02.01 behaves over the point $\mathfrak{p}$: unramified primes correspond to étale morphisms, ramified primes to non-étale ones, and the decomposition group is the stabiliser inside the étale fundamental group. The discriminant-different formula is the coherent-sheaf avatar of the conductor of the cover.

• Riemann surfaces and the analytic notion of ramification 06.02.01. Unit 06.02.01 treats the topological and complex-analytic ramification of a holomorphic map between Riemann surfaces — a covering branched at isolated points, measured by a local index $e$. The ramification here is the algebraic-integer shadow of that same geometric branching: a prime ideal ramifies when the map $\mathrm{Spec}{\mathcal{O}}_L\to \mathrm{Spec}{\mathcal{O}}_K$ is branched over $\mathfrak{p}$, and the ramification index $e(\mathfrak{P}/\mathfrak{p})$ is the exact arithmetic analogue of the analytic branching order, but read off the factorisation of a prime ideal rather than the local degree of a map of surfaces.

• Local fields and analytic number theory 21.02.07, 21.03.03. Completing at $\mathfrak{P}$ turns $D_{\mathfrak{P}}$ into $\mathrm{Gal}(L_{\mathfrak{P}}/K_{\mathfrak{p}})$, the setting of 21.02.07, where the ramification-group filtration and the Hasse-Arf theorem govern wild ramification and lead into local class field theory. The Frobenius elements assembled here are the input to the Chebotarev density theorem of 21.03.03, which equidistributes them over $G$ and yields the analytic class-number formula and all prime-distribution statistics.

Historical & philosophical context Master

The theory of ramification was forced into being by the same crisis that created ideal theory: the failure of unique factorisation in rings of integers. Dedekind, in Supplement XI to the fourth edition of Dirichlet's Vorlesungen über Zahlentheorie (1894) ^{[Dedekind 1894]}, introduced the different (die Differente) and proved that its prime divisors are exactly the ramified primes, and that the discriminant is the norm of the different — the single theorem that unifies this unit. Hilbert's Zahlbericht (1897) ^{[Hilbert 1897]} systematised the Galois-theoretic viewpoint, organising the primes above $\mathfrak{p}$ under the action of $G$ and isolating the decomposition and inertia groups, and setting the programme that class field theory later completed.

Hensel's Theorie der algebraischen Zahlen (1908) ^{[Hensel 1908]} supplied the local viewpoint: study a prime one $p$-adic completion at a time, where the decomposition group becomes the literal Galois group of an extension of complete discretely valued fields. In that local setting the ramification groups of Hilbert, refined by the lower and upper numberings, measure wild ramification; the Hasse-Arf theorem (Arf 1939, building on Hasse) forces the upper-numbering jumps to be integers for abelian extensions, the bridge to local class field theory. Frobenius's 1896 paper ^{[Frobenius 1896]} attached an element of $G$ to each unramified prime and conjectured its equidistribution, proved in full by Chebotarev in 1926. Serre's Corps locaux (1962) ^{[Serre 1962]} gave the definitive modern local, group-theoretic treatment that the Master tier of this unit follows.

Bibliography Master

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  year      = {1894}
}

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  year    = {1897}
}

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}

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}

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}

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  title     = {Algebraic Number Theory},
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}

@book{Marcus1977,
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  author    = {Lang, Serge},
  title     = {Algebraic Number Theory},
  publisher = {Springer-Verlag},
  series    = {Graduate Texts in Mathematics},
  volume    = {110},
  edition   = {2},
  year      = {1994}
}

@book{StewartTall2015,
  author    = {Stewart, Ian and Tall, David},
  title     = {Algebraic Number Theory and Fermat's Last Theorem},
  publisher = {CRC Press},
  edition   = {4},
  year      = {2015}
}