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A classical theorem due to Dedekind states the following:

Let $O_{K}$ be the ring of integers of a number field $K$, and assume $K$ is generated by adjoining the algebraic integer $\alpha$ to $\mathbb{Q}$. Let $f(x) \in \mathbb{Z}[x]$ be the minimal polynomial of $\alpha$.

Then given a prime ideal $\mathfrak{p} \subseteq \mathbb{Z}$, one can deduce the factorization of the ideal $\mathfrak{p}O_{K}$ from the factorization of the polynomial $f(x)$ in $\mathbb{Z}[x]/\mathfrak{p}$ (under the assumption $\mathfrak{p} \nmid [O_K : \mathbb{Z}[\alpha]]$).

See Theorem 1 in this note by Keith Conrad for full details.

My questions are about how generalizable this theorem is. References would be highly appreciated.

  1. Does it hold when I replace:

    • $\mathbb{Z}$ with $R:=\mathbb{F}_q[T]$,
    • $F:=\mathbb{Q}$ with the field of fractions of $R$,
    • $K$ with a finite extension of $F$,
    • $O_K$ with the integral closure of $R$ in $K$,
    • and $\alpha$ with an element from the integral closure of $R$ in the algebraic closure of $F$?

2 . What is the greatest generality in which the above theorem holds? For instance, is it true when $\mathbb{Z}$ is replaced with an arbitrary Dedekind domain $R$?


[EDIT] 3. It seems that the answer to the first two questions is positive when $K$ is a finite separable extension of $F$. What happens when $K/F$ is not separable?

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I have found two different references which answer both my questions in the affirmative. The first is Zariski and Samuel's 'Commutative Algebra I' (Chapter V, p. 317, Thm. 34) and the other is Neukirch's 'Algebraic Number Theory' (Chapter I, p. 47-48, Prop. 8.3). It seems that the only remaining interesting question is: What exactly happens when the field extension is not separable?

Zariski and Samuel describe a theorem of Kummer which applies to integrally closed domains (more general than Dedekind domains), but restricts to the case where the larger ring $R'$ is of the form $R[y]$, while in Dedekind's theorem, $R[y]$ might be a finite-index subgroup of $R'$. In Kummer's theorem, prime ideals are replaced with maximal ideals (in Dedekind domains, those two notions essentially coincide).

Neukirch describes a theorem which applies to Dedekind domains, and gives exactly Dedekind's Theorem when one works with rings of integers.

Dedekind's theorem is a slight strengthening of Kummer's theorem, specialized to the case of number fields. It doesn't follow directly from Kummer's theorem, but it does follow from the theorem described by Neukirch.

Kummer's theorem is the following:

Let $R$ be an integrally closed domain, $K$ its quotient field, $K'$ a finite algebraic extension of $K$, $R'$ the integral closure of $R$ in $K'$. We suppose that there exists an element $y$ of $R'$ such that $R'=R+Ry+\cdots + Ry^{n-1}$ ($n=[K':K]$) ($y$ is then a primitive element of $K$' over $K$). Let $F(Y)$ be the minimal polynomial of $y$ over $K$. ($F(Y)$ necessarily has its coefficients in $R$.) Let $\mathfrak{p}$ be a maximal ideal in $R$; for every polynomial $G(X)$ over $R$, we denote by $\overline{G}(X)$ the polynomial over $R/\mathfrak{p}$ whose coefficients are the $\mathfrak{p}$-residues of the corresponding coefficients of $G$. Let $\overline{F}(X)=\prod_{i=1}^{g} (f_i(X))^{e(i)}$ be the factorization of $\overline{F}(X)$ into distinct irreducible factors $f_i(X)$ over $R/\mathfrak{p}$; for $i=1,\cdots,g$ we denote by $F_i(X)$ a polynomial over $R$ such that $\overline{F_i}(X) = f_i(X)$. Then the ring $R'$ has exactly $g$ maximal ideals $\mathfrak{P}_i$ which lie over $\mathfrak{p}$, and we have $$\mathfrak{P}_i = R'\mathfrak{p}+R'F_i(y).$$ Furthermore we have $$R'\mathfrak{p} =\mathfrak{D}_1 \cap \mathfrak{D}_2 \cdots \cap \mathfrak{D}_g = \mathfrak{D}_1 \cdot\mathfrak{D}_2 \cdots \cdot \mathfrak{D}_g,$$ where $\mathfrak{D}_i = R'\mathfrak{p} + R'(F_i(y))^{e(i)}$.

I have found Neukirch's theorem here. $\mathcal{o}$ stands for a Dedekind domain, $K$ for its fraction field, $\mathcal{O}$ for the integral closure of $\mathcal{o}$ in $L$, a finite separable extension of $K$:

Neukirch

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