# Irreducible polynomials and prime elements

• I
• darksidemath

#### 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 :)

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.

• darksidemath
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.

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

Aren't ideals generated by irreducible polynomials maximal?

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.##