I'm reading this document: http://www.math.northwestern.edu/~len/d70/chap8.pdf and it has this proof about the field automorphism sending roots to roots:
Could someone explain me the part $f(\sigma(x)) = \sigma(f(x))$? It says that this is valid since $\sigma$ fixes $F$, but this is not evident to me. It makes sense when $x$ is not a root, because then $f(\sigma(x))$ is simply $f(x)$ which is simply $\sigma(f(x))$ since $\sigma$ doesn't change anything.
Just write out what $f$ is. It's some polynomial with coefficients in $F$, so we have $f(x)=a_nx^n+\dots+a_0$ for some $a_0,\dots,a_n\in F$. We then have $$\sigma(f(x))=\sigma(a_nx^n+\dots+a_0)=\sigma(a_n)\sigma(x)^n+\dots+\sigma(a_0).$$ Since $\sigma$ fixes $F$ and $a_0,\dots,a_n\in F$, this is then equal to $$a_n\sigma(x)^n+\dots+a_0=f(\sigma(x)).$$