Let $L$ be a number field, let $p$ be a prime number, and let $I$ be a ideal of $\mathcal{O}_L$ containing $p$. I am not assuming that $\mathcal{O}_L$ or that $I$ is prime. The quotient ring $\mathcal{O}_L/I$ has a natural structure of $\mathbb{F}_p$-algebra.

Question. Do we have an isomorphism of $\mathbb{F}_p$-algebras $$\mathcal{O}_L/I\simeq \mathbb{F}_p[X]/(f)$$ for some nonzero monic polynomial $f$ ?

I know that the answer if YES in several cases.

1. For example, it is true if $\mathcal{O}_L=\mathbb{Z}[\alpha]$ for some $\alpha$.

Indeed, $I=(\beta,p)$ for some $\beta\in\mathcal{O}_L=\mathbb{Z}[\alpha]$. So $\beta=f(\alpha)$. Replacing $f$ by a gcd of $f$ and $\mu_{\alpha,\mathbb{Q}}$ if necessary, one may assume that $f$ is a divisor of $\mu_{\alpha,\mathbb{Q}}$. Then $\mathcal{O}_L/I\simeq \mathbb{F}_p[X]/(\overline{f})$.

2. If $L=K_1K_2,$ where $\mathcal{O}_{K_i}=\mathbb{Z}[\alpha_i]$, and the discriminants of $K_1$ and $K_2$ are coprime, and $p$ is totally ramified in $K_2$, and $I=\mathfrak{p}_2\mathcal{O}_L.$ where $\mathfrak{p}_2$ is the unique prime ideal of $\mathcal{O}_{K_2}$ lygin above $p$. In this case, one may show that $\mathcal{O}_L/I\simeq \mathbb{F}_p[X]/(\overline{\mu}_{\alpha_1,\mathbb{Q}})$.

Nevertheless, I suspect that I am missing obvious counterexamples...

About 2), i wonder if it is a particular case of 1), so there is a side question:

Side question. If $L=K_1K_2,$ where $\mathcal{O}_{K_i}=\mathbb{Z}[\alpha_i]$, and the discriminants of $K_1$ and $K_2$ are coprime, is $\mathcal{O}_L=\mathbb{Z}[\alpha]$ for some $\alpha$ ? For example, is $\alpha=\alpha_1+\alpha_2$ working ?

Any thoughts ?

Greg

Edit. Clearly, my proof of (1) is false. We cannot replace $f$ by a gcd, since $\mathbb{Z}[X]$ is not a PID. However, (2) is correct (even if I didn't write up the proof here), and anyway, the answer in NO (see the answer and the comment below).

Let $L$ be a number field, let $p$ be a prime number, and let $I$ be a ideal of $\mathcal{O}_L$ containing $p$. I am not assuming that $\mathcal{O}_L$ or that $I$ is prime. The quotient ring $\mathcal{O}_L/I$ has a natural structure of $\mathbb{F}_p$-algebra.

Question. Do we have an isomorphism of $\mathbb{F}_p$-algebras $$\mathcal{O}_L/I\simeq \mathbb{F}_p[X]/(f)$$ for some nonzero monic polynomial $f$ ?

I know that the answer if YES in several cases.

1. For example, it is true if $\mathcal{O}_L=\mathbb{Z}[\alpha]$ for some $\alpha$.

Indeed, since $I$ contains $p$, evaluation at $\alpha$ induces a morphism of $\mathbb{F}_p$-algebras $\mathbb{F}_p[X]\to \mathcal{O}_L/I$. This morphism is surjective since $\mathcal{O}_L=\mathbb{Z}[\alpha]$ . Its kernel is generated by a monic polynomial. Done.

2. If $L=K_1K_2,$ where $\mathcal{O}_{K_i}=\mathbb{Z}[\alpha_i]$, and the discriminants of $K_1$ and $K_2$ are coprime, and $p$ is totally ramified in $K_2$, and $I=\mathfrak{p}_2\mathcal{O}_L.$ where $\mathfrak{p}_2$ is the unique prime ideal of $\mathcal{O}_{K_2}$ lygin above $p$. In this case, one may show that $\mathcal{O}_L/I\simeq \mathbb{F}_p[X]/(\overline{\mu}_{\alpha_1,\mathbb{Q}})$.

Nevertheless, I suspect that I am missing obvious counterexamples...

About 2), i wonder if it is a particular case of 1), so there is a side question:

Side question. If $L=K_1K_2,$ where $\mathcal{O}_{K_i}=\mathbb{Z}[\alpha_i]$, and the discriminants of $K_1$ and $K_2$ are coprime, is $\mathcal{O}_L=\mathbb{Z}[\alpha]$ for some $\alpha$ ? For example, is $\alpha=\alpha_1+\alpha_2$ working ?

Any thoughts ?

Greg

Let $L$ be a number field, let $p$ be a prime number, and let $I$ be a ideal of $\mathcal{O}_L$ containing $p$. I am not assuming that $\mathcal{O}_L$ or that $I$ is prime. The quotient ring $\mathcal{O}_L/I$ has a natural structure of $\mathbb{F}_p$-algebra.

Question. Do we have an isomorphism of $\mathbb{F}_p$-algebras $$\mathcal{O}_L/I\simeq \mathbb{F}_p[X]/(f)$$ for some nonzero monic polynomial $f$ ?

I know that the answer if YES in several cases.

1. For example, it is true if $\mathcal{O}_L=\mathbb{Z}[\alpha]$ for some $\alpha$.

Indeed, $I=(\beta,p)$ for some $\beta\in\mathcal{O}_L=\mathbb{Z}[\alpha]$. So $\beta=f(\alpha)$. Replacing $f$ by a gcd of $f$ and $\mu_{\alpha,\mathbb{Q}}$ if necessary, one may assume that $f$ is a divisor of $\mu_{\alpha,\mathbb{Q}}$. Then $\mathcal{O}_L/I\simeq \mathbb{F}_p[X]/(\overline{f})$.

2. If $L=K_1K_2,$ where $\mathcal{O}_{K_i}=\mathbb{Z}[\alpha_i]$, and the discriminants of $K_1$ and $K_2$ are coprime, and $p$ is totally ramified in $K_2$, and $I=\mathfrak{p}_2\mathcal{O}_L.$ where $\mathfrak{p}_2$ is the unique prime ideal of $\mathcal{O}_{K_2}$ lygin above $p$. In this case, one may show that $\mathcal{O}_L/I\simeq \mathbb{F}_p[X]/(\overline{\mu}_{\alpha_1,\mathbb{Q}})$.

Nevertheless, I suspect that I am missing obvious counterexamples...

About 2), i wonder if it is a particular case of 1), so there is a side question:

Side question. If $L=K_1K_2,$ where $\mathcal{O}_{K_i}=\mathbb{Z}[\alpha_i]$, and the discriminants of $K_1$ and $K_2$ are coprime, is $\mathcal{O}_L=\mathbb{Z}[\alpha]$ for some $\alpha$ ? For example, is $\alpha=\alpha_1+\alpha_2$ working ?

Any thoughts ?

Greg

Let $L$ be a number field, let $p$ be a prime number, and let $I$ be a ideal of $\mathcal{O}_L$ containing $p$. I am not assuming that $\mathcal{O}_L$ or that $I$ is prime. The quotient ring $\mathcal{O}_L/I$ has a natural structure of $\mathbb{F}_p$-algebra.

Question. Do we have an isomorphism of $\mathbb{F}_p$-algebras $$\mathcal{O}_L/I\simeq \mathbb{F}_p[X]/(f)$$ for some nonzero monic polynomial $f$ ?

I know that the answer if YES in several cases.

1. For example, it is true if $\mathcal{O}_L=\mathbb{Z}[\alpha]$ for some $\alpha$.

Indeed, $I=(\beta,p)$ for some $\beta\in\mathcal{O}_L=\mathbb{Z}[\alpha]$. So $\beta=f(\alpha)$. Replacing $f$ by a gcd of $f$ and $\mu_{\alpha,\mathbb{Q}}$ if necessary, one may assume that $f$ is a divisor of $\mu_{\alpha,\mathbb{Q}}$. Then $\mathcal{O}_L/I\simeq \mathbb{F}_p[X]/(\overline{f})$.

2. If $L=K_1K_2,$ where $\mathcal{O}_{K_i}=\mathbb{Z}[\alpha_i]$, and the discriminants of $K_1$ and $K_2$ are coprime, and $p$ is totally ramified in $K_2$, and $I=\mathfrak{p}_2\mathcal{O}_L.$ where $\mathfrak{p}_2$ is the unique prime ideal of $\mathcal{O}_{K_2}$ lygin above $p$. In this case, one may show that $\mathcal{O}_L/I\simeq \mathbb{F}_p[X]/(\overline{\mu}_{\alpha_1,\mathbb{Q}})$.

Nevertheless, I suspect that I am missing obvious counterexamples...

About 2), i wonder if it is a particular case of 1), so there is a side question:

Side question. If $L=K_1K_2,$ where $\mathcal{O}_{K_i}=\mathbb{Z}[\alpha_i]$, and the discriminants of $K_1$ and $K_2$ are coprime, is $\mathcal{O}_L=\mathbb{Z}[\alpha]$ for some $\alpha$ ? For example, is $\alpha=\alpha_1+\alpha_2$ working ?

Any thoughts ?

Greg

Edit. Clearly, my proof of (1) is false. We cannot replace $f$ by a gcd, since $\mathbb{Z}[X]$ is not a PID. However, (2) is correct (even if I didn't write up the proof here), and anyway, the answer in NO (see the answer and the comment below).