Prime Ideal Decomposition Ramification and the Different

Choose $0\mathrm{\ne }a\in \mathfrak{a}$. Multiplication by $a$ is an injective endomorphism of the free rank-$n$ lattice ${\mathcal{O}}_K$, with nonzero integral determinant.

Its image $(a)$ has finite index, and $(a)\subseteq \mathfrak{a}\subseteq {\mathcal{O}}_K$; thus ${\mathcal{O}}_K\mathrm{/}\mathfrak{a}$ is a quotient of the finite group ${\mathcal{O}}_K\mathrm{/}(a)$.

Work temporarily with one finite integral basis of $R$ supplied by [F1]; fixing this single existential witness is not a choice from a family. It identifies the additive group of $R$ with ${\mathbb{Z}}^n$, on which fix an explicit natural-number code (the usual sign code and a finite tuple-pairing code). An integral basis is $\mathbb{Q}$-independent: clearing denominators in a rational relation gives an integer relation. Its $[K:\mathbb{Q}]$ elements therefore form a $\mathbb{Q}$-basis of $K$. For any $z\in K$, clearing the finitely many rational coordinates gives a nonzero integer $c$ with $cz\in R$; hence $K=\mathrm{Frac}(R)$. We first prove by induction on $m$ that every subgroup $H\le {\mathbb{Z}}^m$ has a finite basis. The case $m=0$ is immediate. For $m>0$, project onto the first coordinate. By [F2], the image is either zero or $d\mathbb{Z}$ for its least positive element $d$. In the zero case apply induction to the kernel, viewed in the last $m-1$ coordinates. Otherwise take the least-coded $h\in H$ with first coordinate $d$; the kernel has a finite basis by induction, and that basis together with $h$ spans $H$. Their independence follows by taking first coordinates and then using independence in the kernel. This finite induction uses only least natural numbers and one least-coded lift at each nonzero stage.

If $\mathfrak{a}=R$, its factorisation is the empty product. A nonempty finite product of proper prime ideals is contained in any one of its factors, so it cannot equal $R$. Hence this factorisation is unique.

Every ideal $I\subseteq R$ is an additive subgroup of ${\mathbb{Z}}^n$; step 1.1 gives it a finite $\mathbb{Z}$-basis, which also generates it as an $R$-ideal because $\mathbb{Z}\subseteq R$. If $S=R_{\mathfrak{p}}$ and $J\subseteq S$ is an ideal, its contraction $I=J\cap R$ is finitely generated by step 1.1, and $J=IS$: for $r\mathrm{/}s\in J$ with $s\mathrm{\notin }\mathfrak{p}$, the unit $s\mathrm{/}1$ gives $r\mathrm{/}1\in J\cap R$, and the reverse inclusion is immediate. Consequently every ideal of $S$ is finitely generated. Any ascending chain of ideals of $S$ stabilizes: its union is an ideal, hence has finitely many generators. If the union is zero, every member is zero. Otherwise its finite generating list is nonempty, and its generators all lie in one common chain member, which then equals the union. No maximal-condition or dependent-choice implication is used.

Now suppose $0\mathrm{\ne }\mathfrak{a}⊊R$. Put $A=R\mathrm{/}\mathfrak{a}$; [F3] makes $A$ finite. The fixed basis code on $R$ gives each coset of $\mathfrak{a}$ the least code of its representatives. Order the finite set $A$ by these codes. Code each subset of $A$ by its bit mask in this ordering; this gives a definite code to every ideal of $A$. The maximal ideals of $A$ form a finite nonempty set: among its proper ideals, take the least bit-mask code among those of greatest finite cardinality. List all maximal ideals in increasing bit-mask order. By [F4], their inverse images are exactly the maximal ideals of $R$ containing $\mathfrak{a}$. Every nonzero prime containing $\mathfrak{a}$ is on this list: its quotient is finite by [F3] and is a domain, hence a field, so that prime is maximal. Conversely every maximal ideal is prime. The list is therefore exact, finite, and canonically ordered.

Fix a prime $\mathfrak{p}$ in this list and put $S=R_{\mathfrak{p}}$ and $\mathfrak{m}=\mathfrak{p}S$. This is a local domain by [F7], and $\mathfrak{m}\mathrm{\ne }0$ because $\mathfrak{p}$ contains $\mathfrak{a}\mathrm{\ne }0$ and localization of the domain $R$ is injective. By [F5], $R$ is integrally closed; applying [F6] with base domain $R$, field $K=\mathrm{Frac}(R)$ from step 1.1, integral closure $R$ from [F5], and multiplicative set $R\setminus \mathfrak{p}$ shows that $S$ is integrally closed in $K$. Step 1.1 also gives a finite generating list for $\mathfrak{p}$, whose images generate $\mathfrak{m}$.

Code a fraction in $S$ by the least code of a representing pair $(r,s)\in R\times (R\setminus \mathfrak{p})$, using the fixed code on $R$. Thus every nonempty subset of $S$ has a least-coded element. Take the least-coded nonzero $x\in \mathfrak{m}$ and write $x=a\mathrm{/}s$ with $0\mathrm{\ne }a\in \mathfrak{p}$ and $s\mathrm{\notin }\mathfrak{p}$; then $xS=aS$. The map from the localization of $R\mathrm{/}aR$ at the image of $R\setminus \mathfrak{p}$ to $S\mathrm{/}aS$ sending $(r+aR)\mathrm{/}(s+aR)$ to $r\mathrm{/}s+xS$ is a surjective ring homomorphism by [F10]. If its value is zero, then $r\mathrm{/}s\in aS=xS$; since $s\mathrm{/}1$ is a unit, $r\mathrm{/}1\in aS$, and [F8] gives some $t\mathrm{\notin }\mathfrak{p}$ with $tr\in aR$. Thus $(r+aR)\mathrm{/}(s+aR)$ is already zero in that localization, so the map is injective. The source is finite because it is represented by pairs from the two finite sets $R\mathrm{/}aR$ and the image of $R\setminus \mathfrak{p}$, using [F3]. Code its classes by least codes of representing pairs. It is local with maximal ideal $\mathfrak{n}=\mathfrak{m}\mathrm{/}xS$: every class outside $\mathfrak{n}$ has a representative outside $\mathfrak{m}$, hence a unit by [F7]. For any $y\in \mathfrak{n}$, two powers repeat in this finite ring, say $y^i=y^{i+\mathrm{\ell }}$ with $\mathrm{\ell }>0$. Since $1-y^{\mathrm{\ell }}$ is outside $\mathfrak{n}$, it is a unit, so $y^i=0$; necessarily $i>0$, since $1\mathrm{\notin }\mathfrak{n}$. The finite set $\mathfrak{n}$, ordered by these class codes, has a list $y_1,\dots ,y_q$; take for each the least positive nilpotence exponent $k_i$. Every product of $1+{\sum }_i(k_i-1)$ elements of $\mathfrak{n}$ contains some $y_i$ at least $k_i$ times. Thus ${\mathfrak{n}}^N=0$ for some positive $N$, and ${\mathfrak{m}}^N\subseteq xS$.

Take the least positive $N$ with ${\mathfrak{m}}^N\subseteq xS$. If $N=1$, then $\mathfrak{m}=xS$. If $N>1$, choose the least-coded $y\in {\mathfrak{m}}^{N-1}\setminus xS$ and put $z=y\mathrm{/}x\in K$. Then $z\mathrm{\notin }S$ and $z\mathfrak{m}\subseteq S$. If $z\mathfrak{m}\subseteq \mathfrak{m}$, take finite generators $h_1,\dots ,h_q$ of $\mathfrak{m}$ from step 2.3 and take for each $i$ the least-coded coefficient tuple $(a_{ij}{)}_j\in S^q$ with $z{h}_i={\sum }_j{a}_{ij}{h}_j$. The matrix $zI-(a_{ij})$ kills the column of generators. By [F9] its adjugate shows that $\det (zI-(a_{ij}))h_i=0$ for every $i$. Some $h_i\mathrm{\ne }0$, so the domain property gives $\det (zI-(a_{ij}))=0$. The polynomial $\det (TI-(a_{ij}))$ is monic over $S$, making $z$ integral over $S$, contrary to its integral closedness and $z\mathrm{\notin }S$. Thus take the least-coded $h\in \mathfrak{m}$ with $zh\mathrm{\notin }\mathfrak{m}$. Then $zh\in S\setminus \mathfrak{m}$ is a unit by [F7], and for every $t\in \mathfrak{m}$, $t\mathrm{/}h=(zt)\mathrm{/}(zh)\in S$. Hence $\mathfrak{m}=hS$. In either case $\mathfrak{m}=(\pi )$ for a nonzero nonunit $\pi$, chosen by the least codes just specified.

Every nonzero $s\in S$ has a unique expression $s=u{\pi }^k$ with $u$ outside $\mathfrak{m}$ and $k\ge 0$. Divide by $\pi$ whenever the current element lies in $\mathfrak{m}$; the quotient is unique because $S$ is a domain. If division never stopped, the resulting principal ideals would form a strictly ascending chain: equality at a step would make $\pi$ a unit. Step 2.1 rules out such a chain, so division stops at a unit. Uniqueness follows by cancellation: if two exponents differ, a unit would be a multiple of $\pi$, which lies in $\mathfrak{m}$. The recursive divisions are unique, not choices from families.

The ideal $\mathfrak{a}S$ is nonzero and finitely generated by step 2.1. Write each nonzero generator in the form of step 5.1 and take the least of the finitely many exponents, say $e_{\mathfrak{p}}$. Among the finite listed generators attaining that exponent take the least-coded one; it yields ${\pi }^{e_{\mathfrak{p}}}\in \mathfrak{a}S$ after multiplying by a unit inverse; every generator is divisible by that power. Thus $\mathfrak{a}S=({\pi }^{e_{\mathfrak{p}}})={\mathfrak{m}}^{e_{\mathfrak{p}}}$. Since $\mathfrak{a}\subseteq \mathfrak{p}$, this ideal is proper and $e_{\mathfrak{p}}>0$. If $e<f$ and $({\pi }^e)=({\pi }^f)$, cancellation of the nonzero ${\pi }^e$ would make $1$ a multiple of $\pi$, impossible because $\pi \in \mathfrak{m}$. Thus the exponent is unique.

For every $r\ge 0$, multiplication by ${\pi }^r$ induces a surjection $S\mathrm{/}\mathfrak{m}\to {\mathfrak{m}}^r\mathrm{/}{\mathfrak{m}}^{r+1}$; its kernel is exactly $\mathfrak{m}$, since cancellation of the nonzero ${\pi }^r$ shows $c{\pi }^r\in {\pi }^{r+1}S$ exactly when $c\in \pi S$. Thus each layer is one-dimensional over the residue field. Also the maximal ideal of $S\mathrm{/}\mathfrak{a}S$ has nilpotency index exactly $e_{\mathfrak{p}}$: its $e_{\mathfrak{p}}$-th power vanishes, while its preceding power does not, because the powers of $\mathfrak{m}$ are strictly distinct as shown in step 6.1. These local quotient facts support the norm and Dedekind--Kummer uses of the local calculation.

In the canonical finite order of step 2.2 define $\mathfrak{b}={\prod }_{\mathfrak{p}\supseteq \mathfrak{a}}{\mathfrak{p}}^{e_{\mathfrak{p}}}$. This product is nonzero: choose a least-coded nonzero element in each of its finitely many prime factors and multiply them in the domain $R$. At $R_{\mathfrak{p}}$ for a listed prime, every other listed prime becomes the unit ideal (distinct maximal ideals are incomparable), so [step 6.1] gives $\mathfrak{b}{R}_{\mathfrak{p}}=\mathfrak{a}{R}_{\mathfrak{p}}$. At any maximal ideal $\mathfrak{q}$ not on the list, $\mathfrak{a}\subseteq ̸\mathfrak{q}$; take the least-coded element of $\mathfrak{a}\setminus \mathfrak{q}$, which becomes a unit. Each listed prime is distinct from $\mathfrak{q}$, so take its least-coded element outside $\mathfrak{q}$; every factor becomes the unit ideal. Thus the same local equality holds there.

If $x\in \mathfrak{b}\setminus \mathfrak{a}$, its class in the finite ring $R\mathrm{/}\mathfrak{a}$ is nonzero, so its annihilator is proper. The finite, nonempty set of proper ideals containing this annihilator includes the annihilator itself. Take the least bit-mask code among its members of greatest finite cardinality; it is inclusion-maximal among proper ideals, hence a maximal ideal of $R\mathrm{/}\mathfrak{a}$. By [F4], its inverse image is a maximal ideal $\mathfrak{q}$ of $R$ containing $\mathfrak{a}$. The local equality in step 7.2 puts $x\mathrm{/}1$ in $\mathfrak{a}{R}_{\mathfrak{q}}$ by [F8], so take the least-coded $s\mathrm{\notin }\mathfrak{q}$ with $sx\in \mathfrak{a}$. This puts the class of $s$ in the annihilator, a contradiction. Hence $\mathfrak{b}\subseteq \mathfrak{a}$. For the reverse inclusion, $\mathfrak{b}\mathrm{\ne }0$ by step 7.2, so [F3] makes $R\mathrm{/}\mathfrak{b}$ finite; for a hypothetical $x\in \mathfrak{a}\setminus \mathfrak{b}$, its nonzero class has a proper annihilator in $R\mathrm{/}\mathfrak{b}$. The nonempty finite set of proper ideals containing that annihilator includes the annihilator itself. Take the least-bit-mask member of greatest cardinality, using the same quotient coding from step 2.2; it is maximal, and by [F4] its inverse image is a maximal ideal $\mathfrak{q}$ of $R$. The local equality puts $x\mathrm{/}1$ in $\mathfrak{b}{R}_{\mathfrak{q}}$ by [F8], so take the least-coded $s\mathrm{\notin }\mathfrak{q}$ with $sx\in \mathfrak{b}$. Then $s$ lies in the annihilator, a contradiction. Thus $\mathfrak{a}\subseteq \mathfrak{b}$ and $\mathfrak{a}=\mathfrak{b}$.

Suppose a second finite factorisation into distinct nonzero prime ideals is given. Each factor contains $\mathfrak{a}$, so every such prime is maximal by step 2.2. Localizing at any maximal ideal $\mathfrak{p}$, all distinct prime factors become units; the product is proper exactly when $\mathfrak{p}$ occurs, and then its local exponent is the unique exponent of the power of $\mathfrak{p}{R}_{\mathfrak{p}}$. The local equality with $\mathfrak{a}$ therefore forces exactly the same primes and exponents as in step 6.1. The prime list was sorted by least codes, the exponents are unique, and the finite auxiliary choices above were also least-coded. Thus the factorisation is unique and uses no Choice.

Multiplication by $\alpha$ is an injective integer matrix on the integral lattice, and its image is $(\alpha )$.

The index of the image of an injective integer matrix is the absolute determinant; that determinant is the field norm. Hence the quotient cardinality has the asserted value.

Factor both ideals over their finite union of prime supports. The DVR calculation in the factorisation proof makes ${\mathfrak{p}}^r\mathrm{/}{\mathfrak{p}}^{r+1}$ one-dimensional over ${\mathcal{O}}_K\mathrm{/}\mathfrak{p}$, so every successive quotient has cardinality $N\mathfrak{p}$.

Hence $N({\mathfrak{p}}^{r+s})=N({\mathfrak{p}}^r)N({\mathfrak{p}}^s)$ at every prime; multiply over the finite support to obtain the claim.

The finite domain ${\mathcal{O}}_K\mathrm{/}\mathfrak{P}$ is a finite field; its characteristic is a rational prime $p$ and its kernel on $\mathbb{Z}$ is $(p)$.

As a finite-dimensional vector space over ${\mathbb{F}}_p$, that field has $p^f$ elements for some $f\ge 1$, which is exactly $N\mathfrak{P}$.

Localise ${\mathcal{O}}_L$ at $\mathfrak{p}$. It is a finite torsion-free module over the DVR $({\mathcal{O}}_K{)}_{\mathfrak{p}}$, hence free of rank $[L:K]$. Filtering its reduction modulo $\mathfrak{p}$ by the finite prime-power factors, the $e(\mathfrak{P}\mathrm{/}\mathfrak{p})$ layers at $\mathfrak{P}$ are copies of its residue field.

Taking dimensions over ${\mathcal{O}}_K\mathrm{/}\mathfrak{p}$ gives the left side, while the free rank in step 1.1 gives $[L:K]$ for the same quotient.

Substitute the prime factorisation of $\mathfrak{p}{\mathcal{O}}_L$ into its extension to ${\mathcal{O}}_M$ and compare the exponent of $\mathfrak{Q}$.

The three residue fields form a finite tower, so the dimensions multiply. These are exactly the two asserted equalities.

Put $A={\mathcal{O}}_K$, $B={\mathcal{O}}_L$ and $k=A\mathrm{/}\mathfrak{p}$. Monic division by $F$ shows that the kernel of $A[X]\to B$, $X\mapsto \alpha$, is $(F)$: a remainder of degree less than $\mathrm{deg}F$ vanishing at $\alpha$ is zero by minimality over $K$. Monogeneity gives surjectivity, hence $B\mathrm{/}\mathfrak{p}B\simeq k[X]\mathrm{/}(\bar{F})$.

The maximal ideals of this quotient are exactly those generated by the ${\bar{g}}_i$; their inverse images in $B$ are ${\mathfrak{P}}_i=(\mathfrak{p},g_i(\alpha ))$. They are independent of the chosen lifts. Their residue fields are $k[X]\mathrm{/}({\bar{g}}_i)$, of degree $\mathrm{deg}{\bar{g}}_i$, and their contractions are $\mathfrak{p}$. Thus they are exactly the primes above $\mathfrak{p}$.

The local factorisation theorem Integral ideal factorisation in a number field, in ZF writes $\mathfrak{p}B={\prod }_i{\mathfrak{P}}_i^{a_i}$. In the DVR $B_{{\mathfrak{P}}_i}$ used in that theorem's proof, the maximal ideal of $B_{{\mathfrak{P}}_i}\mathrm{/}\mathfrak{p}{B}_{{\mathfrak{P}}_i}$ has nilpotency index exactly $a_i$. On the polynomial side of step 1.1, localisation at $({\bar{g}}_i)$ gives $k[X{]}_{({\bar{g}}_i)}\mathrm{/}({\bar{g}}_i^{e_i})$, whose maximal ideal has nilpotency index exactly $e_i$: its $e_i$-th power vanishes and its $(e_i-1)$-st power does not. Thus $a_i=e_i$, proving the ideal factorisation and the stated ramification and residue degrees.

Let $\mathfrak{P}\mid \mathfrak{p}$ have ramification index $e$. Comparing the valuations of the terms in the Eisenstein equation forces $n{v}_{\mathfrak{P}}(\alpha )=e$; in particular $e\le [L:K]\le n$.

The inequalities in step 1.1 are therefore equalities, so $v_{\mathfrak{P}}(\alpha )=1$ and $e=n$. The resulting factorisation is $\mathfrak{p}{\mathcal{O}}_L=(\mathfrak{p},\alpha {)}^n$; hence this is the unique prime factor and the prime is totally ramified.

Fix an integral basis ${\alpha }_1,\dots ,{\alpha }_n$ as in [F1], and put $A:={\mathcal{O}}_K\mathrm{/}p{\mathcal{O}}_K$. The classes ${\bar{\alpha }}_1,\dots ,{\bar{\alpha }}_n$ form an ${\mathbb{F}}_p$-basis of $A$, since ${\mathcal{O}}_K$ is free over $\mathbb{Z}$ with that basis. Thus ${\dim }_{{\mathbb{F}}_p}A=n\ge 1$, so $p{\mathcal{O}}_K$ is proper; it is nonzero because $p\mathrm{\ne }0$ in the characteristic zero field $K$. By [F5], write $p{\mathcal{O}}_K={\prod }_{i=1}^r{\mathfrak{P}}_i^{e_i}$ with distinct nonzero prime ideals and positive exponents. The list is nonempty because $p{\mathcal{O}}_K$ is proper. Define ${\mathrm{Tr}}_A(a):=\mathrm{tr}(m_a)$ for the ${\mathbb{F}}_p$-linear multiplication map on $A$, and $T_A(x,y):={\mathrm{Tr}}_A(xy)$; matrix-trace linearity makes $T_A$ a bilinear form.

Each ${\mathfrak{P}}_i$ is maximal. Indeed, [F6] makes ${\mathcal{O}}_K\mathrm{/}{\mathfrak{P}}_i$ finite, and [F6] also makes it an integral domain. For any nonzero class, multiplication by it is injective on this finite set, hence surjective, so the class has an inverse and the quotient is a field; [F6] then gives maximality. Therefore distinct ${\mathfrak{P}}_i,{\mathfrak{P}}_j$ are comaximal. Choose $u\in {\mathfrak{P}}_i$ and $v\in {\mathfrak{P}}_j$ with $u+v=1$. In the expansion of $(u+v{)}^{e_i+e_j-1}$, a term $u^k{v}^{e_i+e_j-1-k}$ lies in ${\mathfrak{P}}_i^{e_i}$ if $k\ge e_i$, and otherwise its exponent of $v$ is at least $e_j$, so it lies in ${\mathfrak{P}}_j^{e_j}$. The sum is $1$; hence the powers ${\mathfrak{P}}_i^{e_i}$ and ${\mathfrak{P}}_j^{e_j}$ are comaximal.

Put $A_i:={\mathcal{O}}_K\mathrm{/}{\mathfrak{P}}_i^{e_i}$. Applying [F7] to the pairwise comaximal powers gives $A\cong {\prod }_{i=1}^r{A}_i\mathrm{.}$ If $e_i=1$, then $A_i={\mathcal{O}}_K\mathrm{/}{\mathfrak{P}}_i$ is a finite field. Since ${\mathfrak{P}}_i$ lies above $(p)$, the map $\mathbb{Z}\mathrm{/}p\mathbb{Z}\to A_i$ induced by inclusion of $\mathbb{Z}$ is injective: its kernel is $({\mathfrak{P}}_i\cap \mathbb{Z})\mathrm{/}(p)=0$. Thus $A_i\mathrm{/}{\mathbb{F}}_p$ is a finite field extension, of degree $f_i=f({\mathfrak{P}}_i\mathrm{/}p)$ by definition.

We first prove that the trace pairing of every finite extension $E\mathrm{/}{\mathbb{F}}_p$ of finite fields is nondegenerate, without passing to an algebraic closure. Let $f=[E:{\mathbb{F}}_p]$ and $\sigma (x)=x^p$. By [F8], ${\sigma }^0,\dots ,{\sigma }^{f-1}$ are the distinct automorphisms of $E$ over ${\mathbb{F}}_p$. Fix an ${\mathbb{F}}_p$-basis $b_1,\dots ,b_f$ and form the square matrix $C=({\sigma }^j(b_k){)}_{0\le j<f,1\le k\le f}$. If a row combination with coefficients $c_j\in E$ vanishes, then ${\sum }_j{c}_j{\sigma }^j(b_k)=0$ for each basis vector. Each ${\sigma }^j$ is ${\mathbb{F}}_p$-linear, so ${\sum }_j{c}_j{\sigma }^j(x)=0$ for every $x\in E$, and in particular for every $x\in E^{\times }$. The restrictions ${\sigma }^j{\mathrm{\mid }}_{E^{\times }}$ are distinct group homomorphisms $E^{\times }\to E^{\times }$; [F9] makes them $E$-linearly independent, so every $c_j=0$. Hence $C$ is invertible.

If every $e_i=1$, then [F7] identifies $A$ with the product of the finite fields $A_i$ from step 1.3. Multiplication by $(x_i{y}_i{)}_i$ on this product is block diagonal, so its trace is ${\sum }_i{\mathrm{Tr}}_{A_i\mathrm{/}{\mathbb{F}}_p}(x_i{y}_i)$. The form on $A$ is therefore the orthogonal direct sum of the finite field trace pairings from step 1.4. Each is nondegenerate, so the product pairing is nondegenerate: for a nonzero vector, choose a nonzero component and pair it with a witness in that factor, taking zero in all other factors.

If some $e_i>1$, then ${\mathfrak{P}}_i^{e_i}⊊{\mathfrak{P}}_i$. The containment is immediate; equality would contradict uniqueness in [F5], because the factorizations of ${\mathfrak{P}}_i$ and ${\mathfrak{P}}_i^{e_i}$ have exponents $1$ and $e_i$. Choose $a\in {\mathfrak{P}}_i\setminus {\mathfrak{P}}_i^{e_i}$. Its class in $A_i$ is nonzero and nilpotent, since its $e_i$-th power is zero. Through the product isomorphism in step 1.3, it gives a nonzero nilpotent $\eta \in A$. For any $y\in A$, $u=\eta y$ is nilpotent, so multiplication $T=m_u$ on the finite-dimensional ${\mathbb{F}}_p$-space $A$ is nilpotent.

Steps 2.1 and 2.2 show that the trace pairing on $A$ is nondegenerate exactly when every $e_i=1$. By [F5], this is exactly when $p$ is unramified; equivalently, the pairing is degenerate exactly when $p$ ramifies.

For $z\in {\mathcal{O}}_K$, multiplication by $z$ on ${\mathcal{O}}_K$ has an integer matrix in the integral basis ${\alpha }_1,\dots ,{\alpha }_n$. Viewed over $\mathbb{Q}$, the same matrix represents multiplication by $z$ on $K$, so [F3] makes its matrix trace ${\mathrm{Tr}}_{K\mathrm{/}\mathbb{Q}}(z)$. Reducing the integer matrix modulo $p$ gives multiplication by $\bar{z}$ on $A$; hence ${\mathrm{tr}}_{{\mathbb{F}}_p}(m_{\bar{z}})={\mathrm{Tr}}_{K\mathrm{/}\mathbb{Q}}(z)(\mathrm{m}\mathrm{o}\mathrm{d}p)\mathrm{.}$ Consequently the matrix of $T_A$ in the basis ${\bar{\alpha }}_1,\dots ,{\bar{\alpha }}_n$ is the reduction modulo $p$ of $G=({\mathrm{Tr}}_{K\mathrm{/}\mathbb{Q}}({\alpha }_i{\alpha }_j){)}_{i,j}$. Since each ${\alpha }_i{\alpha }_j\in {\mathcal{O}}_K$ by [F1], [F13] makes every entry of $G$ an integer, and [F4] gives $\det G=d_K$. By [F12], the trace pairing $T_A$ is degenerate exactly when its matrix is not invertible, equivalently when $\det G<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>p=0$; this is equivalent to $p\mid d_K$. Combining with step 3.1 proves that $p$ ramifies exactly when $p\mid d_K$.

The field discriminant is a fixed nonzero integer.

It has only finitely many prime divisors, and the criterion identifies these exactly with the ramified rational primes.

In a $\mathbb{Z}$-basis of the full lattice $A$, nondegeneracy of trace gives a dual basis, so $A^{\vee }$ is again a full lattice and is stable under ${\mathcal{O}}_K$.

The condition $\mathrm{Tr}(xcA)\subseteq \mathbb{Z}$ is equivalent to $cx\in A^{\vee }$, proving $(cA{)}^{\vee }=c^{-1}{A}^{\vee }$ and the fractional-ideal claim.

Lagrange interpolation in the conjugates shows that the trace-dual lattice has $\mathbb{Z}$-basis $\frac{1}{f^{\mathrm{\prime }}(\alpha )},\frac{\alpha }{f^{\mathrm{\prime }}(\alpha )},\dots ,\frac{{\alpha }^{n-1}}{f^{\mathrm{\prime }}(\alpha )}\mathrm{.}$

Thus ${\mathcal{O}}_K^{\vee }=f^{\mathrm{\prime }}(\alpha {)}^{-1}{\mathcal{O}}_K$; taking its fractional-ideal inverse yields $(f^{\mathrm{\prime }}(\alpha ))$.

The dual lattice is obtained from the original basis by the inverse matrix $G^{-1}$, hence its index relative to ${\mathcal{O}}_K$ has absolute determinant $\mathrm{\mid }\det G\mathrm{\mid }$.

Since $\det G=d_K$ and the different is the inverse codifferent, its norm is that same positive index.