Prime ideals in $\mathbb{Z}[x]$ [duplicate]

Question

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• Classification of prime ideals of $\mathbb{Z}[X]$ 1 answer

Am I right that all prime ideals in $\mathbb{Z}[x]$ has the form $p\mathbb{Z}[x]$ for some prime $p\in\mathbb{Z}$?

Thanks a lot!

Answer 1

No, this is not right. There are much more prime ideals. They come in two flavours:

1. Principal ideals $(f)$, where $f$ is either zero, a rational prime, or an irreducible polynomial.
2. Maximal ideals are of the form $(p,f)$, where $p$ is a rational prime, and $f$ is an irreducible polynomial which remains irreducible modulo $p$.

Graphically, here is a "picture" of $\mathrm{Spec}(\mathbb Z[X])$.

Answer 2

You are missing some, for example: $\langle 0 \rangle$ and $\langle x \rangle$ since the quotient is an integral domain