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I would like to find a field $F$ and an epimorphism $\varphi\,\colon \mathbb{Z}_5[X]\to F$ with kernel equal to the ideal generated by the (indecomposable) polynomial $X^2+2$. Is it possible that $F=\mathbb{Z}_5$?

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    Do you know that $K[x]/f(x)$ is a field if $f$ is irreducible over the field $K$?2012-02-23
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    Sure, but I am looking for an explicit epimorphism.2012-02-23
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    What do you mean by an "explicit epimorphism"?2012-02-23
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    A 'concrete' field $F$ together with an epimorphism $\varphi\colon \mathbb{Z}_5[x]\to F$ with kernel equal to $(X^2+1)$.2012-02-23
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    Why isn't $\mathbb Z_5[x]/(x^2+2)$ explicit or concrete? You could also view it as $\mathbb Z_5[\sqrt 3]$2012-02-23
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    What is $\mathbb{Z}_5[\sqrt{3}$?2012-02-23
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    Do you know what is $\mathbb Q[\sqrt{3}]$?2012-02-23
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    Yes, but what this makes no sense for $\mathbb{Z}_5$.2012-02-23
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    Tell us what ${\bf Q}[\sqrt3]$ means to you, and we'll show you that ${\bf Z}_5[\sqrt3]$ makes perfectly good sense.2012-02-23

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