Q. Let $F$ be a field and $f(x) \in F[x]$ be a polynomial of degree $> 0$ over $F$. Show that there is a field $F'$ and an embedding $q:F \to F'$ such that the polynomial $f^q$ has a root in $F'$ where $f^q$ is obtained by replacing each coefficient $a$ of $f$ by $q(a)$.
My attempt: By Kronecker's theorem, there exists a field extension $F'/F$ in which $f(x) \in F[x]$ has a root.
My question: How do I go from here to prove that an field embedding $q$ exists? My guess is that existence of a field extension implies that an embedding should also exist but I'm not sure about it. I also don't see this mentioned anywhere in the definition of a field extension: Wiki
Since, degree of $f(x)>0$, there are only two cases ($f(x)$ cannot be a unit):
(a) $f(x)$ is irreducible
For this case, using Kronecker's theorem we can say that there exists a field extension $F'=F[x]/<f(x)>$ where $f(x)$ has a root. And the mapping given by $q:F \to F'$ where $q(a) = a + <f(x)>$ for $a \in F$ is an injective homomorphism and hence an embedding.
(b) $f(x)$ is reducible
Since $F$ is a field, $F[x]$ is an euclidean domain and therefore a UFD. This implies there exist a product of irreducible polynomials (uniquely up to order and units) for $f(x)$. Let $g(x)$ be any such irreducible polynomial. Then, because any root of $g(x)$ is also a root of $f(x)$ we can apply the same method as in (a) to get an embedding $q(a)=a+<g(x)>$.