I am studying Field Theory from class notes of a senior and need help in deducing an argument.
The argument is as follows: Let $\sigma: K\to L$ be an isomorphism of Fields and let $g$ be an irreducible polynomial in $K[x]$. Then prove that $\sigma g$ is irreducible in $L[x]$.
I tried by arguing that let $\sigma g$ be reducible but I am not able to get a contradiction which I think should be f is reducible.
If $\sigma g = hi$ then $g = (\sigma^{-1}h) (\sigma^{-1}i)$ because the isomorphism $\sigma$ induces isomorphism between their polynomial ring, which is $\sum a_iX^i \mapsto \sum (\sigma a_i)X^i$, and this preserves the degree. contradiction.