Let F⊆K be fields and let f and g be polynomials in F[x]. If f is irreducible in K[x], show that it is irreducible in F[x].

Question

Let $F \subseteq K$ be fields and let $f$ and $g$ be polynomials in $F[x]$. If $f$ is irreducible in $K[x]$, show that it is irreducible in $F[x]$.

Answer 1

Any proper factorization of $f$ in $F[x]$ would be a factorization of $f$ in $K[x]$ since $F \subset K$, but there are none of the latter, so there are none of the former.