Find all prime ideals of $\mathbb{F}_3[x,y]/(y^2−x^3+x)$ whose intersection with $\mathbb{F}_3[x]$ is equal to $(x^2+x+2)$

Question

Find all prime ideals of $\mathbb{F}_3[x,y]/(y^2−x^3+x)$ whose intersection with $\mathbb{F}_3[x]$ is equal to $(x^2+x+2)$.

I'm a completely beginner in commutative algebra and try to learn it in advance. I have actually no idea about how to solve this question.

My thought till now:

1. The prime ideals of $D:=\mathbb{F}_3[x,y]/(y^2−x^3+x)$ are the canonical images of the prime ideals containing $(X^2+Y^2−1)$.

2. $(y^2-x^3+x)$ and $(x^2+x+2)$ are both irreducible in $\mathbb{F}_3[x,y]$; $D$ is integrally closed.

Please give me some hints! I'm really stuck on this problem and lost on the right directions! Thanks very much!

Answer 1

Below is a quite verbose expansion of @LordSharktheUnknown's answer. Though this appears not suited for a beginner in commutative algebra.

$\def\f{\Bbb F}\def\z{\mathbf{Z}}$ We want to find all prime ideals of ${\Bbb F}_3[x,y]/(y^2-x^3+x)$ whose intersection with $\f_3[x]$ is $(x^2+x+2)$.

The elliptic curve $y^2=x^3-x$ means the scheme $\text{Spec}(\f_3[x,y]/(y^2-x^3+x))$. Denote it as $E$. Then the prime ideals in the quotient ring are nothing but points of $E$ over $\f_3$. Denote the residue field of $p$ as $\kappa_p$, i.e. $$\kappa_p=\f_3[x,y]/p.$$

Now consider the maps: $$\f_3[x]\rightarrow\f_3[x,y]\rightarrow\f_3[x,y]/(y^2-x^3+x).$$

And for every point $p$ on $E$ over $\f_3$, consider the composite map: $$\f_3[x]\rightarrow\f_3[x,y]/(y^2-x^3+x)\rightarrow\f_3[x,y]/p=\kappa_p.$$ If the intersection of $p$ with $\f_3[x]$ is $(x^2+x+2)$, then this map has kernel $(x^2+x+2)$, and hence induces the map $$\f_3[x]/(x^2+x+2)\hookrightarrow\kappa_p.$$

But this ring $\f_3[x]/(x^2+x+2)$ is isomorphic to $\f_9$, as the polynomial $x^2+x+2$ is irreducible over $\f_3$, thus the residue field $\kappa_p$ is an extension of $\f_9$. That is to say, $p$ is a point of $E$ defined over $\f_9$ if and only if the intersection of $p$ with $\f_3[x]$ is $(x^2+x+2)$.

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It is a standard fact that the ring $\f_3[x,y]/(y^2-x^3+x)$ is a Dedekind domain, so every ideal can be uniquely factored into products of prime ideals. Therefore we are looking for the prime ideal factorisation of the ideal $(x^2+x+2)$.

By the fact that every ideal in a Dedekind domain is generated by at most two elements, we can write $p$ as $(x^2+x+2,f(x,y))$ for some polynomial $f$.

Now $p$ contains $(y^2-x^3+x)$ means that $y-x^3+x$ can be written as a linear combination of $x^2+x+2$ and $f$: $y-x^3+x=\alpha(x^2+x+2)+\beta f$ for some $\alpha$ and $\beta$.

A calculation shows that $x^3-x=(x+1)^2\pmod{x^2+x+2}$. So we find that $f=y-x-1$ and $f=y+x+1$ satisfy the condition, and they give all the prime ideals we want to find.

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Hope this helps.

Answer 2

In effect you are looking for points on the elliptic curve $y^2=x^3-x$ over the field $\Bbb F_9$. Let $\alpha_1$ and $\alpha_2$ be the solutions of $x^2+x+2=0$ in $\Bbb F_9$. Consider $\beta_i=\alpha_i^3-\alpha_i$.

Two possibilities: (i) the $\beta_i$ are squares in $\Bbb F_9$, (ii) the $\beta_i$ are non-squares in $\Bbb F_9$.

In (i) then $\beta_i=f(\beta_i)^2$ in $\Bbb F_9$ for some polynomial $f$ with integer coefficients, and the ideals you seek are $(x^2+x+2,y\pm f(x))$.

In (ii) then $(x^2+x+2)$ is already prime.

I'll leave you to do the required finite field calculation.