# Describe explicitly $\text{spec}(\mathbb{R}[x])$ and $\text{spec}(\mathbb{C}[x])$ [closed]

Describe explicitly $$\text{spec}(\mathbb{R}[x])$$ and $$\text{spec}(\mathbb{C}[x])$$, (where for a given ring $$R$$, $$\text{spec}(R)$$ is defined to be the set of all prime ideals of $$R$$).

I don't have an attampt or something to start with.

## closed as off-topic by user26857, Riccardo.Alestra, Javi, clathratus, StrantsMar 20 at 21:07

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – user26857, Riccardo.Alestra, Javi, clathratus, Strants
If this question can be reworded to fit the rules in the help center, please edit the question.

• You can start with the definition of $\operatorname{Spec}$. – lisyarus Mar 19 at 17:21
• For a given ring $R$, $\text{spec}(R)$ is defined to be the set of all prime ideals of $R$. @lisyarus – J. Doe Mar 19 at 17:22

## 1 Answer

Let $$F = \mathbb R$$ or $$\mathbb C$$. Since $$F$$ is a field, $$F[x]$$ is a principal ideal domain. What can you say about the prime ideals in a principal ideal domain?

• $F[x]$ is a PID thus each prine ideal is generated by an irreducible polynom, thanks. – J. Doe Mar 19 at 17:36