Primes in an extension over Dedekind domain

Question

I am considering Dedekind domain $\mathbb{Z}[a]/(a^3-a-1)$. I consider the splitting of $23\in \mathbb{Z}$. I know from Daniel Marcas, that if $Q$ is a prime dividing $23$ in the extension, them $Q\cap \mathbb{Z}=(23)$. I need to prove $(23,a-3)$ is a prime dividing $23$, in $\mathbb{Z}[a]/(a^3-a-1)$. Clearly, $(23,a-3)\cap \mathbb{Z}=(23)$. So, does it proves that it is a prime over $23$?

Answer 1

You need to prove that $(23,a-3)$ is a proper ideal (ie. not containing $1$). Then $(23,a-3)\cap \Bbb{Z}$ is a proper ideal of $\Bbb{Z}$ containing $23$ so it has to be $(23)$.

The proof will follow from that $\mathbb{Z}[a]/(a^3-a-1)/(23,a-3)\cong \Bbb{F}_{23}[x]/(x^3-x-1,x-3)\cong \Bbb{F}_{23}[x]/(x-3)\cong \Bbb{F}_{23}$. The latter is a field thus $(23,a-3)$ was a maximal ideal.