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Let $P(x)\in \mathbb{Z}[x]$ be an irreducible polynomial of degree $3$. Suppose that $\alpha_1, \alpha_2, \alpha_3$ are roots of $P(x)$. For what such $P(x)$ is it the case that the ring of integers of $\mathbb{Q}[\alpha_1]$ is $\mathbb{Z}[\alpha_1]$.

Also, let $p$ be a prime, and consider $\mathbb{Z}[\alpha_1]/(p)$. Let $$\text {Nm}(x + y\alpha_1 + z\alpha_1^2) = \prod_{i\le 3}(x + y\alpha_i + z\alpha_i^2)$$

For what primes $p$ is it the case that if $\text{Nm}(x + y\alpha_1 + z\alpha_1^2) = 0$ in $\mathbb{Z}[\alpha_1]/(p)$, then $p|x, y, z$?

I will also will be fine with a method for generating such polynomials and primes.

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    $\begingroup$ I don't think there's a simple answer to the first question. Chapter 7.3 of Alaca and Williams, Introductory Algebraic Number Theory, will give you some idea of the complications involved. $\endgroup$
    – Gerry Myerson
    Commented Aug 8, 2014 at 5:55
  • $\begingroup$ I am also fine with a method for generating such polynomials and primes, as seen in my new edit. $\endgroup$
    – Mayank Pandey
    Commented Aug 8, 2014 at 6:41
  • $\begingroup$ In that case, if $m$ is squarefree, and $m^2\not\equiv1\pmod9$, then the ring of integers in ${\bf Q}(\root3\of m)$ is ${\bf Z}[\root3\of m]$. $\endgroup$
    – Gerry Myerson
    Commented Aug 8, 2014 at 6:45
  • $\begingroup$ Is there any good method for testing whether a polynomial has this property, or whether it satisfies the second property? $\endgroup$
    – Mayank Pandey
    Commented Aug 8, 2014 at 6:52
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    $\begingroup$ Have you gone through Chapter 7.3 of Alaca and Williams yet? $\endgroup$
    – Gerry Myerson
    Commented Aug 8, 2014 at 7:05

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