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 $F=\mathbb{Z}_5$?
Epimorphism onto a field with kernel isomorphic to $(X^2+2)$
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abstract-algebra
polynomials
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0Do 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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0Sure, but I am looking for an explicit epimorphism. – 2012-02-23
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0What do you mean by an "explicit epimorphism"? – 2012-02-23
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0A '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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4Why 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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0What is $\mathbb{Z}_5[\sqrt{3}$? – 2012-02-23
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0Do you know what is $\mathbb Q[\sqrt{3}]$? – 2012-02-23
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0Yes, but what this makes no sense for $\mathbb{Z}_5$. – 2012-02-23
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2Tell 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