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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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    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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it's really very simple: add an "extra element" (let's call it $a$) to $\mathbb{Z}_5$ with the following property: $a^2 = 3$. this gives us the following 25 elements:

$\{0,1,2,3,4,a,1+a,2+a,3+a,4+a,2a,1+2a,2+2a,3+2a,4+2a,3a,1+3a,2+3a,3+3a,4+3a,4a,1+4a,2+4a,3+4a,4+4a\}$

the epimorphism you seek is given by $\varphi(f(x)) = f(a)$ (the "evaluation map"). for example, if $f(x) = x^3 + x + 1$, then $\varphi(f(x)) = a^3 + a + 1 = (a^2)a + a + 1 = 3a + a + 1 = 1 + 4a$