The discussion centers on the relationship between prime elements in the ring Z[√3] and the irreducibility of the polynomial x²−3 in the polynomial ring Fp[x]. It is established that a prime p in Z is a prime element of Z[√3] if and only if x²−3 is irreducible in Fp[x]. The irreducibility of the polynomial ensures that the quotient ring Zp[x]/(x²−3) forms an integral domain, confirming the prime status of p. Additionally, the conversation touches on the distinction between integral domains and fields, particularly regarding ideals generated by irreducible polynomials.
PREREQUISITESMathematicians, algebraists, and students studying abstract algebra, particularly those interested in ring theory and polynomial irreducibility.
I thought the quotient by the ideal generated by an irreducible is a field, not just an integral domain.fresh_42 said: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'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.##WWGD said:Aren't ideals generated by irreducible polynomials maximal?