Basic Field Theory Remarks



A fundamental question of field theory is, given any $f \in F[X]$ does there exists a field ext $K/F$ such that $f(x) = 0$ for some $x \in K$?

The only ideals of a field are $(0)$ and $F$, from which it follows that for $\varphi : F \to F'$ a hom of rings (= fields) is either $\operatorname{im} \varphi = (0)$ or $\approx F$.



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