Existence of a Field Extension Containing a Root


Theorem.  Let $F$ be a field & $f \in F[X]$ an irreducible polynomial.  Then there exists a field ext $K$ containing an isomorphic copy of $F$ in which $f$ has a root. 

Proof.  Consider the ideal $(f) \subset F[X]$.  Since $F[X]$ is a PID $(f)$ is maximal since $f$ is irred.  Let $K = F[X]/(f)$ which is a field since $(f)$ is maximal.  Let $\pi : F[X] \to K$ be the natural projection $\pi(g) = g + (f) \equiv \overline{g}$.  Then $f(\overline{x}) = \overline{f(x)} = 0_K \pmod{(f)}$.  Further, $\pi |_F$ embeds $F$ into $K$ as an isomorphic copy. 

Identifying $F$ with the isomorphic copy, there essentially exists $K/F$ in which $f(x) = 0$ for some $x \in K$.

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