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



Quotients of a ring of integers
The answer to your main question is no. Let $I = (p)$ where $p$ *splits completely* in $L$ and $r := [L:\mathbf Q] > p$. (Example: cubic field in which $p = 2$ splits completely.) Then $\mathcal O\_L/(p) \cong \mathbf F\_p^r$ but if $\mathcal O\_L/(p) \cong \mathbf F\_p[X]/(f)$ for monic $f$ in $\mathbf F\_p[X]$ then $f$ is a product of $r$ distinct monic linear factors, which is impossible for $r > p$. This proves $\mathcal O\_L$ is not $\mathbf Z[\alpha]$ for some $\alpha$ ($L$ is not monogenic).


Dedekind found the first example of this: $L = \mathbf Q(\theta)$ where $\theta$ is a root of $X^3-X^2-2X-8$. This is a cubic field in which $2$ splits completely. Thus $\mathcal O\_L/(2) \cong \mathbf F\_2^3$ and $\mathcal O\_L/(2) \not\cong \mathbf F\_2[X]/(f)$ for any $f$ in $\mathbf F\_2[X]$. Also $\mathcal O\_L \not= \mathbf Z[\alpha]$ for some $\alpha$ in $\mathcal O\_L$.


