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- 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 :)

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

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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.

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

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Aren't ideals generated by irreducible polynomials maximal?

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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.##Aren't ideals generated by irreducible polynomials maximal?

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