I have the following question:
Let $F$ be a field and $K$ an extension field of $F$ of degree $n$. Let $f(x)\in F[x]$ be an irreducible polynomial of degree $m$. Suppose $n$ and $m$ are relatively prime. Show that $f(x)$ is irreducible as a polynomial in $K[x].$
Here is my solution:
Let $u$ be a root of $f(x)$ contained in some field extension of $K.$ From the hypothesis, we have $[K:F]=n$ and $[F(u):F]=m.$ Now, \begin{align} [K(u):K][K:F]&= [K(u):F]\\ &=[K(u):F(u)][F(u):F], \end{align} which implies that $[F(u):F] \mid [K(u):K][K:F]$ i.e. $m \mid [K(u):K]n.$ Since $(m,n)=1,$ we have that $m \mid [K(u):K].$ Hence, $[K(u):K]\geq m.$
However, the degree of $u$ over $K$ is at most the degree of $u$ over $F,$ i.e. $[K(u):K]\leq m.$
Hence, $[K(u):K]= m,$ and therefore $f(x)$ must be irreducible over $K.$
Is this correct and complete solution? If no, what is missing? Thanks in advance!