Prime Ideals of $\Bbb Z[x]$ containing $\mathbb(3, x^2+3x+5)$

Asked Apr 03 '24 at 15:28

Active Apr 06 '24 at 06:08

Viewed 45 times

I need to find the prime ideals of $\mathbb{Z}[x]/(3, x^2+3x+5) \cong \mathbb{Z}_3[x]/(x^2+3x+5) \cong \mathbb{Z}_3[x]/(x^2+2)$.

Therefore, we have that $(x+2)$ and $(x+1)$ are the two maximal ideals of $\mathbb{Z}_3[x]$ containing $(x^2+3x+5)$.

How do I get back the required ideals in $\mathbb{Z}[x]$? What map is to be used to get the answer?

A similar question was asked here. However, it did not answer my question on how to get back the ideals in $\mathbb{Z}[x]$.

edited Apr 06 '24 at 06:08

user26857

asked Apr 03 '24 at 15:28

Anon12

The "required ideals" in $\Bbb Z[x]$ are $\pi^{-1}((x-\varepsilon))=(x-\varepsilon,3)$ ($\epsilon=\pm1$). – Anne Bauval Apr 03 '24 at 15:46
$(3,x+1)$ and $(3,x+2)$ – Shean Apr 03 '24 at 15:46
1

Please don't answer questions in the comments! These are great answers, even if brief, but the question remains open because it has no answers. – Sammy Black Apr 03 '24 at 15:59
Thank you, Anne. The ideal correspondence answered my question. – Anon12 Apr 03 '24 at 16:17

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