The polynom is still irreducible

[mathmari]

Hey!

Let $f(x) \in K[x]$ an irreducible polynom of $K[x]$ of degree $n$.
Let $K\leq F$ a field extension with $[F:K]=m$. If $(n,m)=1$ show that $f(x)$ stays irreducible and as a polynom in $F[x]$.
Could you give me some hints how to show this?? (Wondering)

 

[jvicens]

Hello! To show that $f(x)$ remains irreducible in $F[x]$, you can use the fact that an irreducible polynomial in $K[x]$ remains irreducible in any field extension of $K$. This is because if $f(x)$ can be factored in $F[x]$, then it can also be factored in $K[x]$ by simply considering the coefficients in $K$.

To show that $f(x)$ remains as a polynomial in $F[x]$, you can use the fact that the degree of $f(x)$ in $K[x]$ is $n$, and since $(n,m)=1$, this means that $n$ and $m$ are relatively prime. Therefore, the only way for $f(x)$ to have a factor of degree $m$ in $F[x]$ is if it also has a factor of degree $n$ in $F[x]$, which would contradict the irreducibility of $f(x)$.

Hope this helps! Let me know if you need any further clarification.