Irreducible polynomials and prime elements

[darksidemath]

TL;DR Summary

How can I show that p is a prime element of Z[√3]?

let p∈Z a prime how can I show that p is a prime element of Z[√3] if and only if the polynomial x^2−3 is irreducible in Fp[x]? ideas or everything is well accepted :)

 

[fresh_42]

If $p\in R$ is prime, then $R/(p)$ is an integral domain.

If $x^2-3\in \mathbb{Z}_p[x]$ is irreducible, then $\mathbb{Z}_p[x]/(x^2-3)\cong \mathbb{Z}_p[\sqrt{3}]\cong \mathbb{Z}[\sqrt{3}]/(p)$ is an integral domain.

 

[WWGD]

If $p\in R$ is prime, then $R/(p)$ is an integral domain.

If $x^2-3\in \mathbb{Z}_p[x]$ is irreducible, then $\mathbb{Z}_p[x]/(x^2-3)\cong \mathbb{Z}_p[\sqrt{3}]\cong \mathbb{Z}[\sqrt{3}]/(p)$ is an integral domain.

I thought the quotient by the ideal generated by an irreducible is a field, not just an integral domain.

 

[fresh_42]

It's a field if the ideal is maximal. There are no problems in principal ideal domains, but the general case is more complicated.

 

[WWGD]

Aren't ideals generated by irreducible polynomials maximal?

 

[fresh_42]

Aren't ideals generated by irreducible polynomials maximal?

I'm not sure and have been too lazy to think about it. E.g. we could have a situation $(p) \subsetneq (p,q) \subsetneq R.$