Return to Question

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 $R$?

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?

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 $R$ satisfying the property $|R / \mathfrak{p} | < \infty$ for all prime ideals $\mathfrak{p}$?

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 $R$?