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Let $p \in \mathbb{Z}$ be prime. I am trying to determine all prime ideals $\mathfrak{q} \subset \mathbb{Z}[i]$ such that $\mathfrak{q} \cap \mathbb{Z} = p\mathbb{Z}$. I know that

$$ \{\text{prime ideals }\mathfrak{q} \subset \mathbb{Z}[i] \text{ s.t. } \mathfrak{q} \cap \mathbb{Z} = p\mathbb{Z}\} $$

is bijective to

$$ \{\text{prime ideals in } \mathbb{Z}[i] \otimes \mathbb{Z}/p\mathbb{Z} \cong \mathbb{Z}[X]/(X^2+1) \otimes \mathbb{Z}/p\mathbb{Z} \cong \mathbb{Z}/p\mathbb{Z}[X]/(X^2+1)\}. $$

But how can explicitely determine all $\mathfrak{q}$?

fish_monster
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  • How would you determine the prime ideals of $\mathbb{F}_p[X]/(X^2+1)$? – Aphelli Nov 13 '21 at 09:51
  • @Mindlack I do not know. – fish_monster Nov 13 '21 at 09:57
  • Prime ideals (ee the duplicate):
    1. $(p)$ is a prime ideal if $p\equiv 3\pmod 4$ is a rational prime
    2. $(p)=(\pi_1)(\pi_2)$ with conjugate (and distinct) prime ideals $(\pi_1)$ and $(\pi_2)$ if $p\equiv 1\pmod 4$
    3. $(2)=(1+i)^2$ is the square of a prime ideal.
     – Dietrich Burde Nov 13 '21 at 10:00

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